On Functoriality of Homological Mirror Symmetry of Elliptic Curves ¨ t fu ¨ r Mathematik und Informatik Der Fakulta ¨ t Leipzig der Universita eingereichte
Dissertation zur Erlangung des akademischen Grades
Doctor rerum naturalium (Dr. rer. nat.) im Fachgebiet
Mathematik vorgelegt von
Hilke Reiter geboren am
30. Januar 1976
in
Hoya
Leipzig, 15. Februar 2007
Contents 1 Introduction 1.1 A broader context . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 12 14 15
2 Homological Mirror Symmetry of the Elliptic Curve 2.1 The mirror manifolds . . . . . . . . . . . . . . . . . . . 2.2 Description of the derived category . . . . . . . . . . . 2.3 Description of the Fukaya category . . . . . . . . . . . 2.4 The equivalence . . . . . . . . . . . . . . . . . . . . . .
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Functoriality of the Mirror Functor
31
3 Extensions of the Mirror Functor 3.1 The trivial example of an extension of the mirror functor 3.2 New categories . . . . . . . . . . . . . . . . . . . . . . . . ¡ ¢ 4 Automorphisms of τ, Db (Eτ ) in pDb ell 4.1 The shiftfunctor . . . . . . . . . . . . . . . . . . . . . . . 4.2 The translations . . . . . . . . . . . . . . . . . . . . . . . 4.3 Tensoring with a line bundle . . . . . . . . . . . . . . . . 4.4 Dualizing . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Involution . . . . . . . . . . . . . . . . . . . . . . . . . . .
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37 37 37 38 40 41
5 Isogenies of Elliptic Curves 5.1 The functors πr∗ and πr∗ . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The functors γs∗ and γs∗ . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The symplectic counterpart of γ . . . . . . . . . . . . . . . . . . . .
43 45 46 52
6 Automorphisms of the Elliptic Curve 6.1 The functors corresponding to τ 7→ τ + 1 . . . . . . . . . 6.1.1 The pushforward and its symplectic counterpart 6.1.2 The pullback and its symplectic counterpart . . . 6.2 The functors corresponding to τ 7→ − τ1 . . . . . . . . . . 6.2.1 The pullback and its symplectic counterpart . . . 6.2.2 The pushforward and its symplectic counterpart
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4
CONTENTS
7 Functors Induced by Symplectic Morphisms 97 7.1 The functors corresponding to n : (x, y) 7→ (−y, x) . . . . . . . . . . 103 8 More Functors Between Derived Categories of Elliptic Curves 8.1 Tensoring with a coherent sheaf . . . . . . . . . . . . . . . . . . . . . 8.1.1 Tensoring with a vector bundle . . . . . . . . . . . . . . . . . 8.1.2 Tensoring with a torsion sheaf . . . . . . . . . . . . . . . . . 8.2 Functors related to correspondences . . . . . . . . . . . . . . . . . . 8.3 Functors related to sheaves on the cartesian product Eτ × Eτ 0 . . . . 8.3.1 Locally free sheaves . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Example: The Poincar´e bundle . . . . . . . . . . . . . . . . . 8.3.3 Support in a one dimensional subspace . . . . . . . . . . . . . 8.3.4 Support in a zero dimensional subspace . . . . . . . . . . . . 8.4 A is a nontrivial extension of a torsion sheaf by a vector bundle or a general element of Db (Eτ × Eτ 0 ) . . . . . . . . . . . . . . . . . . . .
111 112 112 117 119 120 121 124 128 132
9 More Functors Between Fukaya Categories 9.1 Functors induced by triples (f, γ, V) . . . . . . . . . . . . . . . . 9.2 Lagrangian correspondences . . . . . . . . . . . . . . . . . . . . . 9.3 Functors induced by Lagrangian correspondences - An approach 9.3.1 Nondegenerate Lagrangian correspondences . . . . . . . . 9.3.2 Degenerate Lagrangian correspondences . . . . . . . . . .
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137 137 139 144 145 146
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10 Synopsis 10.1 Summary . . . . . . . . . . . . . . 10.2 Outlook . . . . . . . . . . . . . . . 10.2.1 Families of elliptic curves . 10.2.2 Lagrangian correspondences
Appendix
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157
A Theta-Functions 157 A.1 The Riemann theta functions . . . . . . . . . . . . . . . . . . . . . . 157 A.2 The meromorphic functions with two periods . . . . . . . . . . . . . 161 B Homological Algebra 165 B.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 B.2 A∞ -categories with transversal structure . . . . . . . . . . . . . . . . 166 C Intersection Theory 169 C.1 Divisors and line bundles on Riemannian manifolds . . . . . . . . . . 170 C.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 D Not yet a mirror functor for families of elliptic curves 177 D.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 D.2 Preliminary thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
CONTENTS
5
List of Symbols
185
Bibliography
189
6
CONTENTS
Chapter 1
Introduction In this thesis we study functorial properties of the mirror functor for the elliptic curve. Kontsevich conjectured in his talk [49] at the ICM 1994 in Z¨ urich that mirror symmetry can best be stated and proofed as the equivalence of two categories: the derived category of coherent sheaves on one manifold of the mirror pair and the Fukaya category on the other. He mentioned in his talk the case of the elliptic curve as a positive example. The example was worked out by Polishchuk and Zaslow. In [63] the authors explicitly constructed an equivalence Φτ : Db (Eτ ) − → FK(E τ ) from the derived category of coherent sheaves on the complex torus Eτ = C/(Zτ +Z) and a version of the Fukaya category of the complexified symplectic torus E τ = (R2 /Z2 , τ dx ∧ dy). We will use this functor to attack an extended problem: Given a functor F : Db (Eτ ) − → Db (Eτ 0 ) between two complex tori, we can construct a corresponding symplectic functor by means of the mirror functor. And since Φτ is an equivalence for all τ, we can also do this in the other direction: Given a functor 0 G : FK(E τ ) − → FK(E τ ) we can define an algebraic counterpart of G. Db (Eτ ) F
Φτ
²
Db (Eτ 0 )
Φτ 0
/ FK(E τ ) Â Â Â ΦF ²
/ FK(E τ 0 )
Db (Eτ )
Φτ
Â
ΦG
/ FK(E τ )
 ²
Db (Eτ 0 )
²
Φτ 0
G
/ FK(E τ 0 ).
We use this property of the mirror functor to answer several natural questions: Problem 1: Functoriality with respect to morphisms of the underlying manifolds We ask, whether Φτ behaves functorial with respect to the category of elliptic curves and the category of complexified symplectic tori: A given morphism in the category of complex tori or in the category of complexified symplectic tori induces a functor between the derived categories respectively the Fukaya categories. The question is whether a functor induced by a morphism corresponds to a functor that is itself induced by a morphism or, more generally, by a global geometric construction.
8
Introduction
Under this notion we summarize functors induced by morphisms of the underlying manifolds and functors induced in any way by a geometric object that is fixed for the functor, and their compositions. The answer is unfortunately negative. The mirror functor scrambles the data in the mirror category in an astonishing way. This becomes most apparent in chapter 6 when studying the functors induced by automorphisms of the elliptic curve. In this case already the data of the objects are mixed in a way that cannot be achieved by a geometric construction. Theorem 1.1. Functors between derived categories of elliptic curves induced by morphisms between the underlying complex tori do not in general correspond to functors between Fukaya categories with a global geometric interpretation. Problem 2: Describe the impact of choices on the algebraic side involved in constructing the mirror functor The mirror functor as constructed by Polishchuk and Zaslow depends on various choices. This calls for a description of how the constructed functor depends on these choices. Though every elliptic curve is isomorphic (as a Riemann surface) to a complex torus, the torus is not uniquely determined by the curve. We want to understand the impact of the choice of a specific lattice Z + Zτ such that our given elliptic curve E is isomorphic to C/(Z + Zτ ). Two different choices differ by an automorphism of the elliptic curve, which yields an isomorphism between the two complex tori. Therefore we need to investigate the equivalences induced by these isomorphisms and their symplectic counterparts. For a “good” equivalence Φτ we would expect that the symplectic functor corresponding to these equivalences is again some functor for which we have a geometric explanation in terms of complexified symplectic tori. This is a particular case of what we expect of mirror symmetry - allowing us to build a dictionary of corresponding geometric constructions. But unfortunately, we will see that the mirror functor does not have this expected property. We have no geometric interpretations for these new functors. Theorem 1.2. The construction of the derived category of coherent sheaves and therefore of the mirror functor depends on non-canonical choices. The functors for two such choices differ by equivalences on both the algebraic and the symplectic side. In general there is no geometric interpretation on the symplectic side for these equivalences. Problem 3: Describe the impact of choices on the symplectic side involved in constructing the mirror functor There are similar choices involved in the construction of the Fukaya category. The real part of the complexified symplectic structure on the mirror manifold of Eτ is only defined modulo Z, so that instead of the structure τ dx ∧ dy also (τ + m)dx ∧ dy with m ∈ Z may be chosen. For a proper understanding of the functor FK(E τ ) − → FK(E τ +1 ), we would need it to be induced by an isomorphism of the tori or a comparable geometric construction, which is however not the case. We see that although this question arises naturally on the symplectic side, and
9 the corresponding algebraic functor has an extraordinary simple explanation, the symplectic functor has no intrinsic geometric interpretation. Theorem 1.3. The construction of the Fukaya category depends on noncanonical choices. The categories for two such choices differ by an equivalence. There is no geometric interpretation for these equivalences. New functors between Fukaya categories Finally, this method allows us to construct many new functors between Fukaya categories. We can write down explicitly and explain by geometrical constructions many more algebraic functors than symplectic functors. With the help of the mirror symmetry functor and the resulting new functors on the symplectic side, we can shed more light onto the still quite unknown Fukaya category and its internal symmetries. The common principle: Extension of the mirror functor For the construction of the mirror functor as in [63, 51] we need to fix τ from the beginning. Behind all mentioned motivating questions stands the question: What happens when τ changes? We wish to construct a more general functor Φ between the categories of derived categories and Fukaya categories that specializes for a given τ to Φτ . The following is a rough, not yet mathematical precise outline of what we are going to study: Denote by Db ell the category whose set of objects is the set of all Db (Eτ ) for τ in the upper complex half plane H, its morphisms are a subset of the functors between these categories. In analogy to Db ell denote by FKell the category whose objects are the FK(E τ ) for τ ∈ H where morphisms are functors between the categories. We would like to extend the mirror functor Φτ to a functor Φ : Db ell − → FKell. For objects, we define Db (Eτ ) 7→ FK(E τ ). To establish a functor we need to define how morphisms are mapped, i.e. we have to find for each functor between derived categories F : Db (Eτ ) − → Db (Eτ 0 ) a corresponding functor 0
ΦF : FK(E τ ) − → FK(E τ ). Here we use that the mirror functor is an equivalence and define ΦF to be the functor that makes the following diagram commute: Db (Eτ ) ²
Φτ
F
Db (Eτ 0 )
/ FK(E τ ) ²
Φτ 0
ΦF
/ FK(E τ 0 ).
10
Introduction
Figure 1.1: A topological picture of the Dehn-twist: Cut along the fat line, twist a full turn and glue back.
To solve the above mentioned problem 1, we could consider as morphisms in Db ell and in FKell only those functors that are induced by morphisms of the underlying manifolds, and ask whether Φ is a functor between the resulting categories. For a discussion of problem 2 we could define as morphisms in Db ell the functors induced by isomorphisms and in FKell the functors with geometric interpretation and ask whether Φ is a functor between these categories. Problem 3 is a bit different, since in its statement it does not involve the mirror functor. We ask whether we can intrinsically interpret functors resulting from choices made for FKell and note that this is not the case. Then we actually use the mirror functor to explain those functors, which we already encountered, when studying problem 2. If we define a morphism to be just any functor, Φ becomes by definition a functor and even an equivalence. We are then left with what was described in the motivation: We can describe new functors on both sides by mirror symmetry and natural functors between the mirror categories. Much of this thesis consists of the explicit calculation of functors and their symplectic counterparts. We mention some of them: The shift functor on the algebraic side corresponds to the shift on the symplectic side by definition of the mirror functor. For the same reason, the isogenies πr : Erτ − → Eτ for r ∈ N given by the inclusion of the lattices, correspond to certain r−fold symplectic coverings pr : E rτ − → Eτ . Another beautiful correspondence is that of the Dehn-twist (see figure 1.1) on the symplectic side, and the tensoring a certain fixed line bundle on the algebraic side. On both the complex and the symplectic torus we have an involution, and also in this case the induced functors correspond. These functors are all described in chapter 4, where we treat some automorphisms of the derived category that seem natural and simple. This includes, apart from the above mentioned examples, also the dualizing functor and functors induced by translations. It will turn out, however, that their symplectic counterparts are not induced by a global geometric construction, such as a morphism of the underlying manifolds. Already these elementary examples show that we are much more familiar with the algebraic category than with the Fukaya category, where comparably few functors are known and studied. Functors and equivalences between derived categories have been studied by various people. We know only little about equivalences between derived categories of varieties of high dimension. However, up to dimension three, we understand these equivalences explicitly. Tom Bridgeland [13] proved for the case of dimension three
11 the conjecture of Bondal and Orlov [12] that two birational varieties have equivalent derived categories. Nice survey articles about equivalences of derived categories are written by Bridgeland [15, 14] and as well by Bondal and Orlov [9]. Most relevant for us are previous works by Orlov [59, 60]. These allow to explain geometrically all equivalences Db (Eτ ) − → Db (Eτ 0 ), if Eτ ∼ = Eτ 0 . In chapter 6 we study a different type of autoequivalence of the derived category on an elliptic curve: those induced by changes of the lattice. As mentioned above, a given elliptic curve E corresponds to different lattices which we can choose to be of the form Λτ = Z+Zτ, such that E ∼ = C/Λτ . If τ ∈ H is such that C/Λτ is isomorphic 0 to E, then the lattice Zτ + Z also corresponds to E if and only if there exists an element γ ∈ SL2 (Z) such that τ 0 = γ · τ . The special linear group is generated by the two elements µ ¶ µ ¶ 0 1 1 1 , and , −1 0 0 1 which correspond to τ 0 = − τ1 and τ 0 = τ + 1, respectively. We will hence investigate the functors induced by the isomorphisms Eτ − → Eτ +1 and Eτ − → E− 1 . How could τ a symplectic counterpart of the induced functors look like at all? The symplectic manifolds E τ and E τ +1 have the same real symplectic form, which is given by the imaginary part of the complexified symplectic structure. Therefore, the identity can be interpreted as a symplectic map between them. It will turn out, however, that the identity does not induce a functor on the respective Fukaya categories. 1 The symplectic manifolds E τ and E − τ are in no particular correspondence — there is no morphism between them. Therefore it might not be too surprising that the symplectic counterpart to the change τ 7→ − τ1 scrambles the data of the Fukaya categories in a way that is not represented by a global geometric object. The functor corresponding to τ 7→ τ + 1 is also intrinsically important on the symplectic side, because it describes the change induced by different choices of a lift of the so called B-field. This is piece of the physical data on our manifolds. It is an element in H 2 (T, R)/H 2 (T, Z) and therefore only defined modulo Z. Strangely enough, it will turn out in chapter 7 that we have no intrinsic explanation of the functor induced by the choice. Only by applying the mirror functor a natural interpretation can be given by means of the algebraic functor, corresponding to the described change of base. Another equivalence is induced by the morphism n : (x, y) 7→ (−y, x) between the symplectic tori. We describe this functor in chapter 7. As mentioned above, we know from results of Orlov [59, 60] that the induced autoequivalence of the derived category of the elliptic curve has a global geometric object that induces this functor. In this case, however, this geometric object is no longer a morphism, but a sheaf on the product Eτ × Eτ . Functors of this type are studied in chapter 8. We distinguish the functors induced by such sheaves by the dimension of their support. This construction provides many examples. Especially interesting are the functors induced by sheaves with two dimensional support. If a vectorbundle on Eτ × Eτ 0 induces an equivalence between the respective derived categories, this functor is called a Fourier–Mukai transformation. In the summary in chapter 10 a complete overview over all functors that were studied in this thesis is given.
12
1.1
Introduction
A broader context
Mirror symmetry originates from theoretical physics. Mirror symmetry denotes the phenomenon in string theory that there exist “mirror pairs” of different Calabi-Yau manifolds, giving rise to isomorphic physical theories. The name is derived from a ˆ of dimension symmetry in the Hodge diamond: For dual Calabi-Yau manifolds X, X n (originally only n = 3) one has: ˆ Ωq ). dim H p (X, Ωq ) = dim H n−p (X, From its beginnings in string theory it has also spread to and by now reached the world of mathematics, has inspired research in a lot of different areas and found various formulations only one of which is homological mirror symmetry for CalabiYau manifolds. For a survey of the early history of mirror symmetry especially in physics see Vafa [39, Introduction], and for the development in mathematics see the book of Polishchuk [61, section 6.1.]. For different formulations of early mirror symmetry and its development see the collections of articles edited by Yau [77] and references therein. Mirror symmetry was discovered when physicists studied N = 2 supersymmetry. Dixon [19] and Lerche, Vafa and Warner [52] noticed that there exist pairs of CalabiYau manifolds that lead to the same underlying worldsheet theory. Examples were provided among other by Greene and Plesser [33]. It was conjectured that this may be true in general for Calabi-Yau target spaces. Mathematicians do not yet have a definition of mirror symmetry that covers all examples of mirror manifolds known to physicists. There exist several definitions in different contexts that do not completely agree. An early formulation was given by Batyrev [7] in the context of toric geometry. For an overview over toric mirror symmetry see for example the survey article of Batyrev [8]. Vafa [75] noted that mirror symmetry implies the isomorphism of two cohomology rings, which is used as a mathematical definition of mirror symmetry. Mathematicians were especially attracted to mirror symmetry after Candelas, de la Ossa, Green and Parkes [16] had used mirror symmetry to predict the number of rational curves on a Calabi-Yau manifold. They calculated the number of rational curves on one of the mirror manifolds by solving the PicardFuchs equation, which is a differential equation coming from variations of the Hodge structure, on its mirror partner. A good introduction for mathematicians is written by Morrison [55]. The understanding of this phenomenon improved enormously by the work of Kontsevich and Manin [47] and their view on this question in the general context of Gromov-Witten invariants. By using more elaborate techniques, many of these invariants were computed, confirming the predictions made by physicists. See the articles of Givental [30] and Kontsevich [48]. A new interpretation of the mirror conjecture was proposed by Kontsevich [49] at the International Congress of Mathematicians 1994 in Z¨ urich: “We propose here a conjecture in slightly vague form which should imply the ‘numerical’ Mirror conjecture. Let (V, ω) be a 2n-dimensional symplectic manifold with c1 (V ) = 0 and W a self-dual n-dimensional complex manifold. The derived category constructed from the Fukaya
1.1 A broader context
13
category F (V ) (or a suitably enlarged one) is equivalent to the derived category of coherent sheaves on a complex variety W .” Another important idea was proposed in the famous paper [73] of Strominger, Yau and Zaslow with the title “Mirror symmetry is T-duality”. In this paper they developed a setting that is generally referred to as SYZ-mirror symmetry. They conjectured that mirror dual Calabi-Yau manifolds are fibred over the same base in such a way that generic fibres are dual tori and each fiber of any of these two fibrations is a Lagrangian submanifold. Examples were presented among other by Ruan [64] and Gross [35]. The study of mirror symmetry widened also in other directions: Instead of CalabiYau manifolds, physicists also considered Fano varieties as target spaces. Auroux, Katzarkov and Orlov proved [5] homological mirror symmetry for del Pezzo surfaces. They use a construction developed by Seidel [70] for the Fukaya category of the mirror Landau-Ginzburg model. The example of X = CP2 and its mirror Landau-Ginzburg model was mentioned by Seidel [69] and proved rigourously as a special case of weighted projective spaces in 2004 by Auroux, Katzarkov and Orlov [4]. Seidel developed in [70] and [69] the directed category of Lagrangian vanishing cycles. Instead of describing the whole Fukaya category, he used exceptional collections of objects. His construction already proved powerful. It was used by Auroux et. al. to proof results for Fano varieties as mentioned above. And Seidel himself used his construction to prove homological mirror symmetry for the quartic surface in [68]. There are more aspects of mirror symmetry that we have not yet mentioned, prominently among them the efforts to calculate the cohomology groups of Calabi-Yau manifolds (for a recent result see for example Batyrev and Kreuzer [6]), which is actually one of the key ideas of mirror symmetry. And there are other active fields of research contributing to the understanding of mirror symmetry. We just briefly mention generalized complex geometry, which was invented by Hitchin [38]. Special cases of a generalized complex structures are complex structures and symplectic structures. See also the thesis of Gualtieri [36]. We have not mentioned the whole discussion on differential graded categories, treated for example by Bondal and Kapranov [11], and other developments in homological algebra that contribute especially to homological mirror symmetry. The deeper understanding of Floer (co)homology is also an active field of research that is important to mirror symmetry. For an introduction to Floer theory and its relation to mirror symmetry see the survey of Schwarz [66]. For an survey of more recent developments the reader is referred to Fukaya [26]. There are so many other people from different research areas whose work and names are linked to mirror symmetry that we cannot name them all in this limited space. We return to homological mirror symmetry for Calabi-Yau manifolds: As mentioned above, the homological mirror symmetry conjecture states that the derived category of coherent sheaves on a Calabi-Yau manifold X is equivalent to the deˆ Those who are not familiar with these terms rived Fukaya category on its mirror X. should think of a derived category as a better version of the original category, in the sense that we can do more calculations within the category like constructing mapping cones, kernels and cokernels. A coherent sheaf is a generalized vectorbun-
14
Introduction
dle, whose rank is allowed to jump on subvarieties, and the objects in the Fukaya category are Lagrangian submanifolds that means half dimensional submanifolds where the symplectic structure vanishes, equipped with additional structure. In our case this additional structure will be a local system. The Fukaya category itself is still a subject of active research and in many aspects not completely understood. It was invented by Fukaya in [23]. A nice and precise presentation can be found in the book of Seidel [67] which also treats functors between Fukaya categories. The first proof of an example of homological mirror symmetry was given by Polishchuk and Zaslow in [63], completed by Kreußler in [51]. In both papers an equivalence between the derived category of elliptic curves with the derived Fukaya category is treated. In [62] Polishchuk equips the derived category of coherent sheaves on the elliptic curve with an A∞ structure and proves equivalence to the Fukaya category. The case of abelian varieties in the SYZ-setting is treated in [31] and [54]. It agrees with the homological mirror symmetry conjecture, which was proven in [24] by Fukaya. As mentioned above, Seidel proved homological mirror symmetry for the quartic surface in [68]. Zaslow used the idea of homological mirror symmetry the other way round to compute the mirror variety by calculations on the Fukaya category in [78] and together with Aldi [2]. Recent efforts are concerned with unifying SYZ mirror symmetry and homological mirror symmetry; see Kontsevich and Soibelman in [50], Fukaya in [25, 22].
1.2
Organization
We will use the mirror functor Φτ : Db (Eτ ) − → FK(E τ ), as defined by Polishchuk, Zaslow [63] and Kreußler [51] throughout this work. Therefore, we give a brief summary of its construction in chapter 2. The core of this work consists of chapters 3 – 9, where we study the possibilities of a functorial extension of the mirror functor. Our setting is rigorously defined in chapter 3. Then we start comparing functors that we consider natural on either the algebraic or symplectic side, with those that we get by using mirror symmetry. First we treat automorphisms of the algebraic category Db (Eτ ) for a fixed τ in chapter 4. This includes the functors induced by shifts, translations, tensoring with line bundles, dualizing, and the involution. We continue with functors on the algebraic side in the next chapters. In chapter 5 and in the following chapter we study those functors that are induced by morphisms of the underlying algebraic manifold (Problem 1). In chapter 6 we concentrate on the functors that arise from different choices of a lattice on the algebraic side and their symplectic counterparts (Problem 2). Then we change our viewpoint to the symplectic side and study functors that arise via morphisms between complexified symplectic tori (related to Problem 2 and solving Problem 3). Until then we will have seen that our definition of natural functor was too restrictive to successfully construct a more general mirror functor. Therefore we extend our notion of morphisms, first in the algebraic category in chapter 8 and then in the symplectic category in chapter 9. Chapter 8 contains several important functors, in particular the Fourier–Mukai transformations. Chapter 9 contains some considerations of how Lagrangian correspondences with local systems induce functors on the corresponding categories. With the end of chapter 9, we have discussed
1.3 Acknowledgements
15
all functors between the given categories that we explicitly know. We end with a summary in chapter 10, where we give another, more detailed overview over all the functors, we have considered. We conclude with an outlook on two open problems that have been touched in our investigations and which would be interesting continuations of the work presented here. Namely this are the construction of a mirror functor for deformations of the elliptic curve and a study of the functors induced by Lagrangian correspondences. The appendix contains mainly material that can be found in textbooks. The exception is appendix D, which contains some thoughts and calculations concerning mirror symmetry for families of elliptic curves.
1.3
Acknowledgements
First I would like to thank both of my advisors Prof. Annette Huber-Klawitter and Prof. Matthias Schwarz for introducing me to the subject and for the support and encouragement they provided. I also want to thank Prof. Bernd Kreußler for the kind help in understanding his article, and Tobias Finis for a helpful discussion about theta functions. Special thanks go to Christian Liedtke for proofreading parts of this work, and to Rainer Munck, who was always willing to listen patiently to my thoughts and often asked the crucial questions. Thanks also to Peter Albers, for many constructive discussions. I am grateful to Matthias Kurzke, Matthias Wendt, Cornelia Schneider, Christoph Sachse and Bj¨orn M¨aurer for serving as test readers. Finally I would like to express my gratefulness to Matthias Kurzke for all the support he provided in the last years and especially in the last months of finishing this thesis.
16
Introduction
Chapter 2
Homological Mirror Symmetry of the Elliptic Curve In this chapter we define and describe the category Db (Eτ ), i.e. the derived category of coherent sheaves on the elliptic curve and FK(E τ ), the Fukaya category of its mirror manifold. In the last section finally the mirror functor Φτ : Db (Eτ ) − → FK(E τ ) is defined. This chapter is a summary of the articles of Polishchuk and Zaslow [63] and Kreußler [51] and contains no complete proofs. For a more detailed study we refer to the original articles. The presentation roughly follows [51].
2.1
The mirror manifolds
Given an arbitrary elliptic curve, there exists a lattice Λ in C , such that C/Λ is as a Riemann surface isomorphic to the given curve and Λ is spanned by the basis {1, τ }, with τ lying in the upper complex half plane. Denote this Riemann-Surface by Eτ . Its mirror manifold is denoted by E τ and is the standard torus R2 /Z2 = C/(Zi + Z) with the complexified symplectic form τ dx ∧ dy. However, for a given elliptic curve, Λ is not uniquely determined, and so we get a family of mirror manifolds, which are not symplectomorphic. On the mirror manifold the complexified symplectic structure is also not uniquely determined. The so called B− field, which enters as the real part of τ, is defined in R/Z, not in R. This problem is studied in detail in chapter 6 and in 9; from now on we fix an element τ and call the associated manifolds Eτ and E τ as the mirror manifolds.
2.2
Description of the derived category
Definition 2.1. A decomposable coherent sheaf is a coherent sheaf that can be written as the direct sum of (at least) two coherent sheafs; a coherent sheaf that is not decomposable is called indecomposable. With this definition the derived category of coherent sheaves has an extraordinarily simple description:
18
HMS for the Elliptic Curve
Proposition 2.2. The full subcategory of the category Db (Eτ ), whose objects are finite direct sums of shifted, indecomposable coherent sheaves on Eτ is equivalent to Db (Eτ ). The main ingredient of the proof is the fact that the homological dimension of the category of coherent sheaves on Eτ is one. This can be deduced from results of Kashiwara and Schapira [41], for details see the article of Kreußler [51]. In the sequel we will only consider this subcategory and still denote it by Db (Eτ );this does no harm, because we are only interested in this category up to equivalence, anyway. Vector bundles over complex tori can be described by multipliers (see for example the book of Griffiths and Harris [34]); we explain this notion for our situation of a one dimensional complex torus. Over C all vector bundles are trivial. Consider a holomorphic vector bundle F over Eτ with fibre V . We have the quotient map π:C− → Eτ = C/Λτ . The pullback of F along π is isomorphic to the trivial bundle. Choose a trivialization f : π ∗ F − → C × V . For z ∈ C, λ ∈ Λτ , the fibres of π ∗ F at z and z +λ are by definition both identified with the fibre of F at π(z), and comparing the biholomorphic trivialization f at z and z + λ yields a linear automorphism of V ∼ = Cr : V o
fz
(π ∗ F)z = Fπ(z) = (π ∗ F)z+λ
fz+λ
3/ V .
mλ
Denote this linear function by mλ . The set {mλ ∈ Aut(V ), λ ∈ Λ} is called a set of multipliers for F. They necessarily fulfill the following relation: mλ0 (z + λ)mλ (z) = mλ (z + λ0 )mλ0 (z) for all λ, λ0 ∈ Λ. Conversely, given a set of multipliers, satisfying this relation we can associate a vector bundle on Eτ by taking it to be the quotient space of C × V under the identification (z, v) ∼ (z + λ, mλ (z)v). Due to the compatibility relation, it suffices to give a multiplier for each element of the basis of Λ. Write Fτ (V, mτ , m1 ) for the vector bundle with fibre V and multipliers mτ in τ −direction and m1 in 1−direction. Usually, our vector bundles will have multiplier m1 = 1; in that case we usually drop the third entry and write Fτ (V, mτ ) for the bundle Fτ (V, mτ , 1). Likewise the subscript τ is dropped, when there is no danger of confusion. The most important bundle for us is the line bundle Lτ (ϕ0 ) = F (C, ϕ0 ), with ϕ0 (z) = ϕ0 (τ, z) = exp(−πiτ − 2πiz). This is a line bundle of degree one. Note that the Chern class and degree agree under the isomorphism H 2 (Eτ , Z) ∼ = Z given by the natural orientation on Eτ as is shown in Griffiths and Harris [34, chapter 1.1.], see remark C.6. The vectorspace of global sections is one dimensional by Riemann-Roch and generated by the theta function θ[0, 0](τ, z) (see appendix A for the definition), which has (as a function of z ∈ C) the zeros 1+τ 1+τ 2 + Zτ + Z and therefore the line bundle corresponds to the divisor ( 2 ) on Eτ . This line bundle generates the whole Picard group, in the sense that any line bundle
2.2 Description of the derived category
19
is given by Lτ (ϕ) = Fτ (C, ϕ) = t∗x Lτ (ϕ0 ) ⊗ Lτ (ϕ0 )n−1 , with ϕ = t∗x ϕ0 · ϕn−1 . Note 0 that t∗x Lτ (ϕ0 ) is the line bundle corresponding to the divisor ( 1+τ − x). To fully 2 describe all the vector bundles over Eτ , we use the following theorem of Atiyah: Theorem 2.3. For every indecomposable vector bundle A over the elliptic curve Eτ , there exist r, n ∈ N, a complex vectorspace V and a nilpotent endomorphism N of V with one dimensional kernel, such that A is isomorphic to the bundle ¡ ¢ πr∗ Lrτ (ϕ) ⊗ Frτ (V, exp N ) , where πr : Erτ −→ Eτ is given by the inclusion of lattices, and ϕ is given by ϕ(z) = t∗x ϕ0 (rτ, z) · ϕ0 (rτ, z)n−1 . Proof. This follows from the classification of vector bundles over the elliptic curve by Atiyah [3, theorems 5 and 7]. ¡ ¢ Definition 2.4. A vector bundle F = πr∗ Lrτ (ϕ) ⊗ Frτ (V, exp N ) , is called preferred. If further r = 1, we call it special. Definition and Proposition 2.5. An indecomposable coherent sheaf S that is not a vector bundle, is a torsion sheaf with support at a single point x. That means, all the fibres of S are zero, only the fibre over x is nontrivial. Denote this fibre by V = Sx . Let z be the usual holomorphic coordinate on C, then OEτ ,x is a local ring with maximal ideal (z − x) and the OEτ ,x -module structure of Sx is given by an endomorphism N of V that describes the action of z − x on the fibre Sx . This data suffices to fully describe S. We will denote such a torsion sheaf by Sτ (x, V, N ). Lemma 2.6. If A ∈ Coh(Eτ ) is a coherent sheaf on the elliptic curve Eτ , it is a direct sum of locally free sheaves and torsion sheaves with support at a single point. Proof. Suppose A is a nontrivial extension of a locally free sheaf F by a torsion sheaf S : 0− →S− →A− →F − → 0. The group of equivalence classes of such A is isomorphic to Ext1 (F, S) and we calculate: Ext1 (F, S) ∼ = Ext1 (OEτ , F ∨ ⊗ S) ∼ = H 1 (Eτ , F ∨ ⊗ S) when α denotes the support of F ∨ ⊗ S then we can write this as ∼ = H 1 (α, F ∨ ⊗ S|α ), because S is a torsion sheaf on a curve, its support and therefore α is zero dimensional. It follows that Ext1 (F, S) is trivial and therefore such an exact sequence splits.
20
HMS for the Elliptic Curve
Remark 2.7. As in the article of Polishchuk and Zaslow [63] we will only treat morphisms of transversal objects here. A pair of indecomposable sheaves is called transversal either if exactly one of them is a torsion sheaf or if they do not have the same degree. The terminology comes from the symplectic analogue, where the objects corresponding to a transversal pair have transversal underlying Lagrangians. We keep in mind though that all definitions, results and calculations can be extended to the whole category, according to the work of Kreußler [51]. We only treat nontransversal objects in a special cases and give the definitions then. We now describe the morphisms of the category Db (Eτ ). To this end first recall that there is the natural isomorphism HomDb (Eτ ) (A1 [m], A2 [n]) ∼ = HomDb (Eτ ) (A1 , A2 [n − m]) ∼ = Extn−m (A1 , A2 ), for A1 , A2 ∈ Coh(Eτ ). Because we deal with coherent sheaves on a curve, these extension groups vanish for n − m ∈ / {0, 1}. Furthermore, by Serre duality we have HomDb (Eτ ) (A1 , A2 [1]) ∼ = HomCoh(Eτ ) (A2 , A1 )∨ . (For the proof of this version of Serre-duality see the article of Kreußler [51, Lemma 2.7].) From this we see that we only have to know the morphisms in Coh(Eτ ), to fully understand the morphisms in the derived category. Notation 2.8. The space of global section of a coherent sheaf or vector bundle A is often denoted by Γ(Eτ , A) or, if the base space is clear from the context, Γ(A). In the following we will also use the notation H 0 (X, A) or, if the base space is clear from the context, H 0 (A) for this space. For special vector bundles we can write down the morphism space in a simple way: Proposition 2.9. Let A1 , A2 be objects in Coh(Eτ ) given by Ai = Lτ (ϕi ) ⊗ Fτ (Vi , exp Ni ), with ϕi = t∗xi ϕ0 · ϕn0 i −1 . If n1 > n2 then the only element in Hom(A1 , A2 ) is the zero morphism. We use the following simplifying notation V = V2 ⊗V1∨ , ϕ = ϕ2 ϕ−1 1 ,N = N2 ⊗ N1∨ . If n2 > n1 then ¡ ¢ 0 Hom(A1 , A2 ) = H 0 (A2 ⊗ A−1 1 ) = H Lτ (ϕ) ⊗ Fτ (V, exp N ) , and there is an isomorphism ¡ ¢ ¡ ¢ νϕ,N : H 0 Lτ (ϕ) ⊗ V −→ H 0 Lτ (ϕ) ⊗ Fτ (V, exp N ) .
For the case n1 ≤ n2 see remark 2.11 below. Notation 2.10. We will use this isomorphism frequently. If all other data is clear from the context, we denote it simply by ν, dropping the subscripts.
2.2 Description of the derived category
21
Proof. Let z be the canonical coordinate function on C. It also serves as ¡ ¢ a coordinate function on Eτ , since Eτ is a quotient of C. An element f ∈ H 0 L(ϕ) is a function 1 d of z. Let D = − 2πi dz . The isomorphism is then given by the formula: νϕ,N (f ⊗ v) :=
exp( DN n )f v
=
dim XV
³
1 k!
−1 2πi
´k
dk N k f v, dz k nk
k=0
(recall that dim ker N = 1.) For details see Polishchuk and Zaslow [63, proposition 2]. Remark 2.11. If n2 < n1 , the bundle A2 ⊗ A−1 1 has no global holomorphic sections; and also there are no (holomorphic) morphisms A1 − → A2 . If n1 = n2 the coherent sheaves A1 , A2 do not form a transversal pair by definition. Of course we nevertheless understand the morphisms between such bundles: If ϕ2 = ϕ1 , then ¡ ¢ Hom(A1 , A2 ) = Hom F (V1 , exp N1 ), F (V2 , exp N2 ) and this space is isomorphic to ∼ → V2 |f ◦ exp N1 = exp N2 ◦ f }. = {f : V1 − If ϕ1 6= ϕ2 then there are no morphisms A1 − → A2 . Remark 2.12. Formally, we can write N d exp(− 2πin dz )f (z) = f (z −
N 2πin ).
N The expression is defined by inserting z · 1 − 2πin in the powerseries expansion of f .
Let A1 , A2 be the vector bundles over Eτ given by ¡ ¢ Ai = πri ∗ L(ϕi ) ⊗ F (Vi , exp Ni ) =: πri ∗ Ei , ϕi (z) = t∗xi ϕ0 (ri τ, z)ϕ0 (ri , z)ni −1 . Let d = gcd(r1 , r2 ) and r =
r1 r2 d
= lcm(r1 , r2 ). Define π ˜i by the diagram
E = Er1 τ ×Eτ Er2 τ π ˜2
π ˜1
²
Er2 τ
/ Er τ 1 ²
πr2
πr1
/ Eτ
Lemma 2.13. E is isomorphic to Erτ × Z/dZ. Proof. We give an explicit map j and show that it is an isomorphism: j : Erτ × Z/dZ − → Er1 τ ×Eτ Er2 τ ([z]rτ , µ) 7→ ([z]r1 τ , [z + µτ ]r2 τ ) Note that this map is welldefined, since the inclusion of lattices is welldefined. We first show injectivity, then surjectivity.
22
HMS for the Elliptic Curve 1. Injectivity: let ([z]r1 τ , [z + µ0 τ ]r2 τ ) = ([z 0 ]r1 τ , [z 0 + µ00 τ ]r2 τ )
(2.2.1)
and define a, b, a0 , b0 by z = arτ + b, z 0 = a0 rτ + b0 . The following equations for m, n ∈ Z follow from 2.2.1: (a) ar20 = a0 r20 + m (b) ar10 = a0 r10 +
µ r2
+ n with µ = µ00 − µ0
(c) b = b0 mod Z To fulfill these equations, we need mr10 − nr20 =
µ , d
But this equation has no solution for µ 6= 0, since then µd ∈ / Z. For µ = 0 is follows that m is a multiple of r20 and n of r10 , because r10 , r20 have no common divisor. It follows that z = z 0 mod Zrτ + Z. 2. Surjectivity: Let ([z], [z 0 ]) be an element of E. We have to show, it lies in the image of j. From ([z], [z 0 ]) ∈ Er1 τ ×Eτ Er2 τ it follows by definition that z = z 0 in Eτ . From this we get the equations: (a) b = b0 mod Z (b) ar1 = a0 r2 + m for an m ∈ Z; but m = µ + kd for a k ∈ Z and via this we see: ([z], [z 0 ]) = j([z], µ).
Now define π ˜1,µ :
Erτ × {µ}
π ˜2,µ :
Erτ × {µ}
j
/ Er τ ×E Er τ τ 1 2
π1
/ Er τ . 1
j
/ Er τ ×E Er τ τ 1 2
π2
/ Er τ 2
The isomorphism Er1 τ ×Eτ Eτ2 τ ∼ = Z/dZ × Erτ is not canonical, and therefore the precise formula for the π ˜i,µ is not uniquely determined. Instead of the isomorphisms j that we use, we could choose the isomorphism ([z], µ) 7→ ([z + n1 τ ], [z + n2 τ ]) with n2 − n1 = µ, which changes then the π ˜i,µ accordingly. It is immediate that different choices yield isomorphic homomorphism spaces, so that the following proposition does not depend on our choice.
2.2 Description of the derived category
23
Proposition 2.14. With the notation of lemma 2.13, ϕ0 = ϕ0 (ri τ ) and ζ0 = ϕ0 (rτ ); let r10 n2 − r20 n1 > 0. Then we have ³ ¡ ¢ ¡ ¢´ Hom πr1 ∗ L(ϕ1 ) ⊗ F (V1 , exp N1 ) , πr2 ∗ L(ϕ2 ) ⊗ F (V2 , exp N2 ) gcd(r1 ,r2 )−1
∼ =
M
³ ¡ ¢ ∗ ¡ ¢´ ∗ Hom π ˜1,µ L(ϕ1 ) ⊗ F (V1 , exp N1 ) , π ˜2,µ L(ϕ2 ) ⊗ F (V2 , exp N2 )
µ=0 gcd(r1 ,r2 )−1
∼ =
M
¡ r0 n −r0 n −1 ¢ H 0 L(t∗(r0 x2 −r0 x1 +n2 r0 µτ ) ζ0 ζ0 1 2 2 1 ) ⊗ (V2 ⊗ V1∨ ). 1
µ=0
2
1
Proof. The first equality follows from flat base change and the adjointness¡ of pull∗ L(ϕ ) ⊗ back and pushforward. For the second equality we need to calculate π ˜i,µ i ¢ F (Vi , exp Ni ) . First observe that ¡ ¢ ¡ ¢ ¡ ¢ ∗ ∗ π ˜i,µ L(ϕi ) ⊗ F (Vi , exp Ni ) = π ˜i,µ L(ϕi ) ⊗ F Vi , exp(rj0 Ni ) , ¡ ¢ ∗ L(ϕ ) . We with j = 2 if i = 1 and vice versa. So we only have to determine π ˜i,µ i do this by working with the corresponding divisors i h i h D(ϕi ) = 12 + 12 ri τ − xi + (ni − 1) 12 + 12 ri τ . 0
r1 −1h X ¡ ¢ ∗ 1 π ˜2,µ D(ϕ2 ) = 2 + k=0
h
1 2r10 rτ
r10 −1
i − x2 − µτ + kr2 τ +
+
1 2 rτ
r 0 n2 −1
= D(t∗r10 x2 +n2 r10 µτ ζ0 ζ0 1
h
h (n2 − 1) 12 +
k=0
1 2r10 rτ
i − µτ + kr2 τ
i − x2 − µτ + (n2 − + 12 rτ h i h i ∼ 21 + 12 rτ − r10 x2 − n2 r10 µτ + (r10 n2 − 1) 12 + 12 rτ
∼
r10 12
i
X
1)r10 12
)
¡ ¢ ∗ Here we used the propositions C.4 and C.3. For π ˜1,µ D(ϕ1 ) we get an analogous result, without the translation by µτ . This yields: ¡ ¢ ¡ ¢ n r0 −1 ∗ π ˜1,µ (L(ϕ1 ) ⊗ F V1 , exp(N1 ) = L(t∗r0 x1 ζ0 ζ0 1 2 ) ⊗ F V1 , exp(r20 N1 ) 2 ¡ ¢ n r0 −1 ∗ ∗ π ˜2,µ (L(ϕ2 ) ⊗ F (V2 , exp N2 ) = L(tr0 x2 +n2 r0 µτ ζ0 ζ0 2 1 ) ⊗ F V2 , exp(r10 N2 ) , 1
1
and by applying proposition 2.9 we get the desired expression. The composition of morphisms is given by the following formula: Proposition 2.15. Let ϕi = t∗xi ϕ0 ϕn0 i −1 , and let Ni ∈ End(Vi ) as above. Then νϕ1 ,N1 (f1 ⊗ v1 ) ◦ νϕ2 ,N2 (f2 ⊗ v2 ) = ´ ³ n2 N1 −n1 N2 D 1 N2 D )(f ) exp( )(f )(v ⊗ v ) νϕ1 ϕ2 ,N1 +N2 exp( n2 Nn11 −n 1 2 1 2 . +n2 n1 n1 +n2 n2 Or in formal meaning as explained above: ¡ n1 N2 −n2 N1 ν(f1 ⊗ v1 ) ◦ ν(f2 ⊗ v2 ) = ν f1 (z − 2πin )f2 (z + 1 (n1 +n2 )
n2 N1 −n1 N2 2πin2 (n1 −n2 ) )(v1
¢ ⊗ v2 ) ;
the arguments of fi are in End(V2 ⊗ V1∨ ) and to give this a precise meaning one has to tensor with the correct identity matrices.
24
HMS for the Elliptic Curve
2.3
Description of the Fukaya category
The category we will construct here is not the original Fukaya Category, but an adaptation to our situation. We will directly define a category by giving the data of objects and morphisms. The outcome is supposed to be the derived category of the Fukaya category that means, the zeroth homology of the Fukaya category, previously formally extended by formally including mapping cones, but we will not show this. Although this is therefore not entirely true, we will call our category from now on the Fukaya Category and denote it by FK(E τ ). First we define a pre-version of the Fukaya category, denoted F. The objects of this category are triples (Λ, α, M), where Λ is a special Lagrangian of E τ ; in this case this means it comes from a line in C with rational slope. α ∈ R is a logarithm of the slope and M a local system on Λ whose eigenvalues have modulus one, determined by its fibre V and monodromy M . Strictly speaking the monodromy does not determine the local system completely, passing from one orientation to the other changes M to M −1 . But we always choose the orientation of Λ as exp(πiβ), where β is the unique real number − 21 < β ≤ 12 such that β − α ∈ Z. More specifically we write our objects like this: Λ = x(t) + iy(t) ∈ C/Z2 , with x, y linear in t α, such that x(0) + iy(0) + exp(πiα) ∈ Λ M = (V, M ) We define a shift functor on this category: (Λ, α, M)[1] = (Λ, α + 1, M). The morphisms in this category are given by the formula HomF
¡
¢ (Λ1 , α1 , M1 ), (Λ2 , α2 , M2 ) =
( 0 L
if α2 − α1 ∈ / [0, 1) Hom(M | , M | ) if α − α ∈ (0, 1) 1 x 2 x 2 1 x∈Λ1 ∩Λ2
Remark 2.16. Note that Λ1 = Λ2 is equivalent to α2 − α1 ∈ Z and as before we only treat transversal objects (see remark 2.7), which means in this case objects with transversal underlying Lagrangians. There is a natural generalization to the nontransversal case by Kreußler in [51], which gives us a definition of a nontrivial space in the case α2 − α1 = 0. We still have to define the composition of morphisms. Notice that this is the only place, where τ enters in the formulas: To keep notation short we write Li for the triple (Λi , αi , Mi ). We want to define a map: ◦ : Hom(L1 , L2 ) × Hom(L2 , L3 ) − → Hom(L1 , L3 ). So for x1 ∈ Λ1 ∩Λ2 , let u ∈ Hom(M1 |x1 , M2 |x1 ) ⊂ Hom(L1 , L2 ) and for x2 ∈ Λ2 ∩Λ3 , let v ∈ Hom(M2 |x2 , M3 |x2 ). Then u◦v is a tuple with an entry for each intersection point of Λ1 and Λ3 ; the component in Hom(M1 |x3 , M3 |x3 ) is given by the formula Z ³ ´ X exp 2πi h∗ (τ dx ∧ dy) P (M3 ) ◦ v ◦ P (M2 ) ◦ u ◦ P (M1 ). (2.3.1) h
2.3 Description of the Fukaya category z3
25
Λ3
Λ1 x3
z1
K
z2
P (M1 )
?
(u ◦ v)x3
P (M3 )
h(D)
u x1
P (M2 )
-
xK2
v
Λ2
Figure 2.1: Schematic picture of the composition in FK(E τ )
The summation is over all isomorphism classes of orientation preserving holomorphic maps h : D − → E τ , where D is the unit disc in C with three marked points (z3 , z2 , z1 ) on the boundary, ordered anti-clockwise, such that h(zk ) = xk and the connecting arc between zk and zk−1 to Λk . Two such tuples (h, z1 , z2 , z3 ) ˜ z˜1 , z˜2 , z˜3 ) are equivalent, if there is an automorphism f of D such that and (h, ˜ h = h ◦ f and z˜k = f (zk ). We get all such isomorphism classes, by considering the Z−family of triangles as pictured in figure 2.1 and the weight of the sum is then Z exp(2πi h∗ (τ dx ∧ dy) = exp(2πiτ Ah ), where Ah is the area of the triangle corresponding to h. With P (Mk ) we denote the parallel transport on the local system Mk (view figure 2.1): P (Mk ) : Mk |xk−1 − → Mk |xk . Definition 2.17. We now have to enlarge our category to obtain the Fukaya category FK(E τ ). We take formal direct sums of objects in F and define morphisms by matrices. Remark 2.18. On the algebraic side the isogenies πr : Erτ − → Eτ played a crucial role in the description of the vector bundles. There is a symplectic variant of this map: pr : E rτ − → E τ , (x, y) 7→ (rx, y). Define pullback and pushforward as follows: pr∗ (Λ, α, M) = (pr (Λ), α0 , pr∗ M) M ∗ p∗r (Λ, α, M) = (Λ(k) , α00 , (p(k) r ) M), k
26
HMS for the Elliptic Curve
Here α0 and α00 are chosen such that they lie in the same interval (ν − 21 , ν + 12 ) as (k)
α. The Λ(k) are the connected components of p−1 r (Λ), and pr = pr |Λ(k) . For to calculate the mirror functor on morphisms we need a symplectic analogue to lemma 2.13: Lemma 2.19. Given pr1 : E r1 τ −→ E τ and pr2 : E r2 τ −→ E τ we can consider the fibre product p˜2
E := E r1 τ ×E τ E r2 τ p˜1
²
²
E r1 τ Define d = gcd(r1 , r2 ), ri0 = isomorphism is given by
ri d ,r
=
/ E r2 pr2
/ Eτ
pr1
r1 r2 d .
.
E is isomorphic to Z/dZ × E rτ . An explicit
j : Z/dZ × E rτ −→ E r1 τ ×E τ E r2 τ ¡ ¢ ¡ µ, (x, y) 7→ (r20 x, y), (r10 x +
¢ µ r2 , y) .
The proof is just the same as for lemma 2.13. And we also have an analogue of proposition 6.1: Proposition 2.20. For two objects in the Fukaya category pr1 ∗ L1 , pr2 ∗ L2 there is a functorial isomorphism gcd(r1 ,r2 )
Hom(pr1 ∗ L1 , pr2 ∗ L2 ) =
M
Hom(˜ p∗1,ν L1 , p˜∗2,ν L2 ),
ν=1
with p˜i,ν = p˜i ◦ j(ν, ·), with p˜i and j defined as in lemma 2.19. The proof can be found in Kreußler [51].
2.4
The equivalence
In this section we finally define the mirror functor Φτ : Db (Eτ ) − → F K(E τ ). As before we will restrict ourselves to transversal collections of objects. We define Φτ (A[n]) = Φτ (A)[n] and Φτ (⊕Ai ) = ⊕Φτ (Ai ). Now it suffices to define the map for an object A[0], where A is an indecomposable coherent sheaf, and morphisms between these objects. First consider a torsion sheaf S(aτ + b, V, N ) and define Λ = (−a, t), ¡ ¢ Φτ S(aτ + b, V, N ) = (Λ, α, M) with α = 12 , ¡ ¢ M = V, exp(2πib1V + N ) . For special vector bundles we define:
Λ = (a + t, (n − 1)a + nt), ¡ ∗ ¢ Φτ L(taτ +b ϕ0 ϕn−1 ) ⊗ F (V, exp N ) = (Λ, α, M) with α ∈ (− 12 , 21 ), 0 ¡ ¢ M = V, exp(−2πib1V + N ) .
2.4 The equivalence
27
And finally ³ ¡ ³ ¢´ ¡ ¢´ Φτ πr∗ Lτ (ϕ) ⊗ Fτ (V, exp N ) = pr∗ Φrτ Lτ (ϕ) ⊗ Fτ (V, exp N ) . Now we have to define maps: ¡ ¢ Φτ : HomDb (Eτ ) (A1 , A2 ) − → HomFK(E τ ) Φτ (A1 ), Φτ (A2 ) . Depending on the sheaves we will distinguish the following cases: First we consider only special vector bundles, then we generalize this to arbitrary vector bundles and finally we still have to consider morphisms to, from, and between torsion sheaves. For now let us suppose Ai = Lτ (ϕi ) ⊗ Fτ (Vi , exp Ni ),
with ϕi = t∗ai τ +bi ϕ0 ϕn0 i −1 .
Either n2 < n1 and both Hom-spaces are zero, or n1 > n2 and we can apply proposition 2.9 ¡ ¢ HomDb (Eτ ) (A1 , A2 ) ∼ = H 0 L(ϕ) ⊗ V, with
ϕ = t∗(a2 −a1 )τ +(b2 −b1 ) ϕ0 ϕ0n2 −n1 .
¡ ¢ H 0 L(ϕ) is a (n2 − n1 )-dimensional vectorspace. A basis is given by the thetafunctions £ k ¤¡ ¢ , 0 (n2 − n1 )τ, (n2 − n1 )z , 0 ≤ k < n2 − n1 , fk = t∗x2 −x1 θ n2 −n 1 n2 −n1
which is our standard basis for such a space. (For the definition of the theta function and its properties see appendix A.) The images of A1 and A2 under the mirror functor are: ³ ¡ ¢´ Λi = (ai + t, (ni − 1)a + ni t), αi , Mi = Vi , exp(2πibi 1Vi + Ni ) , and by definition we get: ¡ ¢ HomFK(E τ ) Φτ (A1 ), Φτ (A2 ) =
M
Hom(V1 , V2 ) · ek .
ek ∈Λ1 ∩Λ2
A simple calculation modulo Z gives us the intersection points explicitly: ³a − a + k n a − n a + n k´ 2 1 1 2 2 1 1 ek = , with 0 ≤ k < n2 − n1 . n2 − n1 n2 − n1
(2.4.1)
Via these considerations we see that the involved morphism spaces have the same dimension. Essentially we want to map fk to ek , but we will need some correction terms to ensure functoriality. We define: ¡ ¢ fk ⊗ T 7→ C(A1 , A2 )T · ek ∈ Hom (M1 )ek , (M2 )ek . And the correction term C(A1 , A2 ) ∈ End(V2 ⊗ V1∨ ) is calculated to be ³ ´ ³ ¡ ¢´ 2 ∨ b −b1 a2 −a1 1) ∨ 2 ∨ − 1V ⊗ N − 2πi 1 exp −πi( (an22−a ) exp N ⊗ 1 . V ⊗V V 1 2 2 −n1 n2 −n1 2 1 n2 −n1 1 The explicit calculations and the proof that C(A1 , A2 ) establishes functoriality, can be found in [63].
28
HMS for the Elliptic Curve
Remark 2.21. If n2 = n1 our objects are not transversal. The homomorphism space is zero if the support of the divisors corresponding to the bundles do not agree. If they do agree, this is a morphisms space between the local systems, described in remark 2.11. On the symplectic side, if n2 = n1 the underlying Lagrangians either do not intersect at all, or are equal. By choosing the right definition (see [51]) the morphisms spaces are the same on both mirror objects. Definition and Proposition 2.22. Let A1 , A2 be the vector bundles over Eτ given by ¡ ¢ Ai = πri ∗ L(ϕi ) ⊗ F (Vi , exp Ni ) =: πri ∗ Ei , ϕi (z) = t∗xi ϕ0 (ri τ, z)ϕ0 (ri , z)ni −1 . Let d = gcd(r1 , r2 ), ri0 =
ri d
r1 r2 d .
and r =
Define π ˜i by the diagram π ˜1
E = Er1 τ ×Eτ Er2 τ π ˜2
/ Er τ 1
²
²
Er2 τ
πr1
/ Eτ
πr2
E is isomorphic to Erτ × Z/dZ. For each ν ∈ Z/dZ we defined above π ˜i,ν : Edr10 r20 τ − → Eri τ , where π ˜i,ν is given by the inclusion of lattices, composed with the identity on Er2 and the translation by ντ on Er1 τ . (See proposition 2.14 and the preceding lemma.) We can then define Φτ by the following diagram Hom(πr1 ∗ E1 , πr2 ∗ E2 )
∼
/
L ν
Hom(˜ πr∗1 ,ν E1 , π ˜r∗2 ,ν E2 ) L
Φτ
²
L
∼ =
¡ ¢ Hom pr1 ∗ Φr1 τ (E1 ), pr2 ∗ Φr2 τ (E2 )
Φrτ
¡ ¢ πr∗1 ,ν E1 ), Φrτ (˜ πr∗2 ,ν E2 ) ν Hom Φrτ (˜
¡ ¢ Hom Φτ (πr1 ∗ E), Φτ (πr2 ∗ E2 ) ²
²
∼
/
L ν
²
∼ =
¡ ¢ Hom p˜∗r1 ,ν Φr2 (E1 ), p˜∗r2 ,ν Φr2 (E2 ) (2.4.2)
If A1 is a torsion sheaf and A2 a vector bundle, both Hom-spaces, on the algebraic side and on the symplectic side between the symplectic counterparts are zero. If A2 is a torsion sheaf, A2 = S(a2 τ + b2 , V2 , N2 ), then let A1 = L(ϕ1 ) ⊗ F (V1 , exp N1 ) a vector bundle, then there is a canonical isomorphism ¡ ¢ Hom(A1 , A2 ) ∼ = V2 ⊗ V1∨ = Hom Φτ (A1 ), Φτ (A2 ) , since the symplectic counterparts only intersect in one point. The isomorphism Φτ : V2 ⊗ V1∨ − → V2 ⊗ V1∨
2.4 The equivalence
29
is then given by the matrix ¡ ¢ f · exp −N2 ⊗ (a1 + na2 )1V1∨ + 1V2 ⊗ a2 N1∨ , with ¡ ¢ f = exp −πiτ (na22 + 2a1 a2 ) − 2πi(a2 b1 + a1 b2 + na2 b2 ) This definition can be extended to A2 = πr∗ E1 , with E1 = L(ϕ1 ) ⊗ F (V1 , exp N1 ) by the commutative diagram Hom(E1 , πr∗2 A2 )
Φrτ
∼ = adjoint prop.
² Hom(πr∗ E1 , A2 )
Φτ
³ ¡ ¢´ / Hom Φrτ (E1 ), Φrτ πr∗ (A2 )
∼ =
∼ =
² ³ ¡ ´ ¢ / Hom Φτ πr∗ (E1 ) , Φτ (A2 )
¡ ¢ / Hom Φrτ (Ee1 ), p∗r Φτ (A2 ) ∼ = adjoint prop.
∼ =
² ³ ¡ ´ ¢ / Hom pr∗ Φrτ (E1 ) , Φτ (A2 )
Here we used among other things the identity Φrτ (πr∗ A2 ) = p∗r Φτ A2 . (Compare this with the results of section 5.1.) Theorem 2.23. Φτ : Db (Eτ ) −→ FK(E τ ) is an equivalence of categories. For the proof of this theorem one first has to show that map between the morphism spaces was defined such that Φτ is a functor. This is the hardest work, but was already done by choosing the right correction terms C(A1 , A2 ). That it is an isomorphism is obvious, since we explicitly defined it on respective bases of the involved vectorspaces of morphisms. What is then left to show is that for each object L in FK(E τ ) there is an object A in Db (Eτ ) such that Φτ (A) is isomorphic to L. The proof uses the fact, that we can decompose all our objects into direct sums, such that the summands already have the correct form, so we can see immediately what A should be. Definition 2.24. Because Φτ is an equivalence, for each object L in FK(E τ ) there exists an object A in Db (Eτ ), such that Φτ (A) is isomorphic to L. A itself is isomorphic to a direct sum of shifted torsion sheaves and preferred vector bundles. We call A the algebraic counterpart of L. Analogously we define the symplectic counterpart of A as the object L in FK(E τ ) such that Φτ (A) = L.
30
HMS for the Elliptic Curve
Chapter 3
Extensions of the Mirror Functor A functor between two derived categories of coherent sheaves over an elliptic curve, corresponds via the mirror construction to a functor between Fukaya categories of complexified symplectic tori. In the following chapters we investigate this correspondence. We can easily construct functors between derived categories by global geometric constructions, such as for example morphisms of the underlying elliptic curves. It will turn out, that these correspond to functors between Fukaya categories that are rather artificial in that they lack a geometric interpretation. In chapter 2 we have defined the functor Φτ : Db (Eτ ) − → FK(E τ ) for a fixed τ . Since all formulas we have discussed so far depend explicitly on τ , it is a natural question to ask, what happens when we vary τ . To vary τ in the upper complex half plane H means we are looking at the tautological family of elliptic curves E − → H (see definition of families of elliptic curves in appendix D). This is not a moduli space, since different τ correspond to the same elliptic curve. Namely Eτ ∼ = Eτ 0 if and only if there is a γ ∈ SL2 (Z) with µ ¶ a b γ= , c d
and τ 0 = γ · τ =
aτ + b . cτ + d
We now consider the derived categories of coherent sheaves in each fibre. The set of the derived categories of the fibres contains all derived categories of elliptic curves over C. But to a given derived category in this set, there correspond several fibres. It is a surprising result of Bondal and Orlov[10] that we can reconstruct the elliptic curve from its derived category. So the ambiguity that is left, is the choice of a lattice, as described above, or, in other words: To a given derived category of coherent sheaves over an elliptic curve, corresponds a SL2 (Z) orbit {SL2 (Z) · τ } in H and the corresponding quotients C/(Zγ · τ + Z), γ ∈ SL2 (Z), which themselves correspond to a unique elliptic curve. A γ ∈ SL2 (Z) also defines an isomorphism of the corresponding fibres: µ ¶ z a b γ= , Eτ − → Eγτ , z 7→ . c d cτ + d
32
Extensions of the Mirror Functor
Note that γ and −γ induce the same transformation of the lattice, while they induce different morphisms. All automorphisms of an elliptic curve can be given by an SL2 (Z)-induced map between two different quotients corresponding to the same elliptic curve. A morphism of elliptic curves induces via pullback and pushforward functors between the derived categories. In the case of isomorphisms, these functors are equivalences. Given a functor F between two such derived categories D1 , D2 , together with the information, over which τ ∈ H we are operating, we can construct a functor ΦF between the corresponding Fukaya categories. (We need to specify τ , so we can pick the correct mirror object out of the family of corresponding mirror manifolds and the correct mirror functor.): D1 = Db (Eτ ) F
²
D2 = Db (Eτ 0 )
Φτ
Φτ 0
/ FK(E τ ) ²
ΦF
/ FK(E τ 0 )
Constructing functors between derived categories is fairly easy, and we know more functors than just those induced by morphisms of elliptic curves: Let X, Y be two smooth projective varieties, and A an object of Db (X × Y ). Denote by pX and pY the projections onto the first, respective the second factor. Then A defines the two functors ¡ L ¢ ΦA : Db (X) − → Db (Y ), ΦA (·) = RpY ∗ A ⊗ p∗X (·) ¡ L ¢ ΨA : Db (Y ) − → Db (X), ΨA (·) = RpX∗ A ⊗ p∗Y (·) . Results of Orlov in [59] and [60] show, how powerful this construction is: Theorem 3.1. 1. Let X, Y be smooth projective varieties. Suppose that F : Db (X) −→ Db (Y ) is an exact functor and an equivalence of triangulated categories. Then there is a unique (up to isomorphism) object A ∈ Db (X × Y ) such that the functor F is isomorphic to the functor ΦA . 2. If X, Y are abelian varieties, then A has only one non-trivial cohomology, that is, A is isomorphic to the object B[n], where B is a sheaf on X × Y.
3.1
The trivial example of an extension of the mirror functor
If we want to extend our functor Φτ : Db (Eτ ) − → FK(E τ ) to a functor Φ which specializes to Φτ for a given τ , this needs to be a functor from derived categories to Fukaya categories, so we have to construct a functor of 2-categories. A first step is to define these 2-categories. First we define the 2-category of derived categories of elliptic curves pDb ellall , the p standing for “pairs”. We need to fix τ to define Φτ (see the previous chapter). The choice of τ is inherent to our notation but given Db (Eτ ) as an abstract category, we cannot reconstruct τ. Therefore we take objects in the 2-category pDb ellall to be
3.1 The trivial example of an extension of the mirror functor
33
pairs (τ, D) of an element τ of H and a category D such that D = Db (Eτ ). (That is to say, fix such an equivalence and use the subcategory and the representatives for the isomorphism classes of coherent ¡ sheaves, needed ¢ to construct Φτ as described in chapter 2.) As morphisms Hom (τ1 , D1 ), (τ2 , D2 ) choose the category of all functors F : D1 − → D2 between the derived categories and remember τ1 , τ2 , such that a morphism is a triple (τ1 , τ2 , F ). We need to distinguish F : (τ1 , D1 ) − → (τ2 , D2 ) from F 0 : (τ10 , D1 ) − → (τ20 , D2 ), also in the case where Eτi0 ∼ = Eτi , for to satisfy axiom 1 of the definition of categories B.1. Furthermore we need this information to apply the mirror functors correctly, as mentioned above. With morphisms between morphisms defined to be natural transformations. Similarly we define the 2-category pFKellall of Fukaya categories with the objects being pairs (τ, F) of an element of H and a Fukaya category F, such that F = FK(E τ ). Morphisms are triples (τ1 , τ2 , F ), with F : FK(E τ1 ) − → FK(E τ2 ) a functor, morphisms between morphisms are natural transformations. This is a classical construction, and it is clear that these really are 2-categories. Most of the time, the data of the base points τ ∈ H is clear from our notation and we simple write FK(E τ ), Db (Eτ ) for objects in these categories and F for a morphism. To define Φ, we define for objects ¡ ¢ ¡ ¢ τ, Db (Eτ ) 7→ τ, FK(E τ ) . And for morphisms F : (τ1 , D1 ) − → (τ2 , D2 ) we define ΦF : (τ1 , ΦD1 ) − → (τ2 , ΦD2 ), via the commutative diagram of functors: D1 = Db (Eτ ) F
²
D2 = Db (Eτ 0 )
Φτ
Φτ 0
/ FK(E τ ) ²
ΦF
/ FK(E τ 0 )
For a natural transformation σ : F1 − → F2 of functors Fi : Db (Eτ ) − → Db (Eτ 0 ), 0 τ we define the natural transformation Φσ of functors ΦFi : FK(E ) − → F K(E τ ) as follows: σ being a natural transformation means, for each A ∈ Db (Eτ ) we are given a Db (Eτ )−morphism σA : F1 A − → F2 A, such that for each morphism ˜ f ∈ HomDb (Eτ ) (A, A), the following diagram commutes: F1 A F1 f
²
F1 A˜
σA
σA˜
/ F2 A ²
F2 f
/F A ˜ 2
Let now L be an object in FK(E τ ). Then there is an object A ∈ Db (Eτ ),¡ such ¢that Φτ (A) is isomorphic to L, since Φτ is an equivalence. And ΦFi L = Φτ 0 Fi (A) by definition of ΦFi . With this notation define (Φσ)L := Φτ 0 (σA ).
34
Extensions of the Mirror Functor
The necessary commutativity of the diagram ΦF1 L (ΦF1 )f
²
˜ ΦF1 L
(Φσ)L
(Φσ)A˜
/ ΦF2 L ²
(ΦF2 )f
/ ΦF L ˜ 2
follows from the definition of ΦFi , the fact that Φτ and Φτ 0 are equivalences and the corresponding property of σ. So we successfully extended our mirror functor to the tautological family of elliptic curves. Obviously Φ : pDb ellall − → pFKellall is again an equivalence. But it is not a very satisfactory construction, because it is only formal and we have not gained any insight so far. What we would like to do, is to compare functors between derived categories and Fukaya categories more directly – especially some natural functors. This will be done in the forthcoming chapters.
3.2
New categories
We define a new category pDb ell as subcategory of pDb ellall : The ¡ objects ¢in this new category are the same as before. That means they are pairs τ, Db (Eτ ) , with τ ∈ H. Morphisms between them are triples (τ1 , τ2 , F ), where F is a functor and a composition of the following pure cases: → Eτ1 a holomorphic map of Riemann surfaces, 1. the pullback f ∗ for f : Eτ2 − 2. the pushforward f∗ for f : Eτ1 − → Eτ2 a holomorphic map of Riemann surfaces, 3. the shift functor [n], n ∈ Z, 4. F = · ⊗ L, the functor that tensors with a fixed line bundle L on Eτ1 = Eτ2 , and 5. the dualizing functor F 7→ RHom(F, OEτ1 ) = F ∨ with τ1 = τ2 . Remark 3.2. We could summarize numbers one and two by considering functors corresponding to a correspondence f . For now, we treat this as two different cases. Correspondences are studied in section 8.2. ¡ ¢ Define further the category pFKell for which objects are pairs τ, FK(E τ ) , and morphisms are pairs (τ1 , τ2 , G) that are compositions of the following pure cases: → E τ1 , 1. the pullback g ∗ for g : E τ2 − 0
2. the pushforward g∗ for g : E τ − → E τ , and 3. the shift functor [n]. The definition of pFKell is not complete, because we have not specified, what 0 → E τ should be. Natural choices would be: g : Eτ − 1. a complexified symplectic map (i.e. respects the complexified symplectic structure),
3.2 New categories
35
2. a real symplectic map, i.e. respects only the imaginary part of the complexified symplectic structure (this actually does not define functors on the Fukaya category; see 7.4) or 0
3. a correspondence D ⊂ E τ × E τ , maybe equipped with a local system. Remark 3.3. The third choice is very promising, however, we will not be able to define a category of comlexified tori with them. This is because of problems with transversality that will be discussed in section 9.2. The first two choices define a category of complexified symplectic tori. It will turn out in chapter 7 that the morphisms of type 2 do not induce functors between the respective Fukaya categories and therefore are not a suitable class of morphisms for us. Since it is not much effort discussing all three possibilities in the following chapters, we will not exclude them at this point from our considerations. There are even more functors for which we have an interpretation than those we have included here. In chapters 8 and 9 we discuss further extensions of these categories. Remark 3.4. As above the categories pDb ell and pFKell naturally ¡ carry ¢ the struc0 ture of a 2-category, where morphisms between morphisms Hom F, F are natural transformations from F to F 0 . Now we check, whether we can extend Φτ to a functor Φ : pDb ell − → pFKell. Define for objects ¡ ¢ ¡ ¢ Φ : τ, Db (Eτ ) 7→ τ, FK(E τ ) , and for morphisms we need to define ΦF for each F as the functor that makes the diagram commute Db (Eτ ) ²
Φτ
F
Db (Eτ 0 )
Φτ 0
/ FK(E τ ) ²
ΦF
/ FK(E τ 0 ).
Definition 3.5. Consider a functor F : Db (Eτ ) − → Db (Eτ 0 ) between derived categories of coherent sheaves on an elliptic curve and define G = ΦF as before. We then call G symplectic counterpart of F and F algebraic counterpart of G. Have we now really extended Φτ ? For to proof that this is a functor, we first have to investigate the different (pure) cases of morphisms on the algebraic side and check, whether the formally defined functor ΦF lies within the category pFKell. If ΦF always exists, then this naturally commutes with composition of functors and composition of natural transformation of functors and Φ is a functor. To proof that it is an equivalence, we have to show that all symplectic morphisms (and objects) have an algebraic counterpart.
36
Extensions of the Mirror Functor
Chapter 4
Automorphisms of pDbell
¡
¢
τ, Db(Eτ ) in
In this chapter we discuss some¢immediate and obvious automorphisms that means ¡ b autoequivalences of τ, D (Eτ ) . There are more automorphisms of this category, though. The functors discussed here are elementary: The shift functor, pullback and pushforward with respect to translations, tensoring with a line bundle, dualizing and pullback and pushforward with respect to the involution z 7→ −z. With the exception of general translations and dualizing, they all have symplectic counterparts.
4.1
The shiftfunctor
The symplectic counterpart of the algebraic shiftfunctor is by definition the symplectic shiftfunctor, and [±1] is a morphism in pFKell. There is nothing left to show.
4.2
The translations
Translations commute with πr , because πr is on the universal covering C just the identity: s
z+ y _
Erτ o
ty
Erτ
πr
²
z+y k
¸ z_
πr
²
Eτ o
ty
²
Eτ
) z²
Therefore also the respective pullbacks and pushforwards commute according to flat base change: ³ ¡ ³ ¡ ¢´ ¢´ n−1 ∗ ∗ t∗y πr∗ L(t∗x ϕ0 ϕn−1 ) ⊗ F (V, exp N ) = π t L(t ϕ ϕ ) ⊗ F (V, exp N ) r∗ y x 0 0 0 ³ ´ = πr∗ L(t∗x+y ϕ0 ϕn−1 ) ⊗ F (V, exp N ) 0
¡ ¢ Automorphisms of τ, Db (Eτ ) in pDb ell
38
On the morphisms, t∗y operates just as the identity: Because we translate both bundles, the translations cancel each other. ¡ ¢ Hom(t∗y A1 , t∗y A2 ) ∼ = H 0 (ty∗ A2 ) ⊗ (t∗y A1 )∨ = H 0 (t∗y−y A2 ⊗ A∨ 1) = H 0 (A2 ⊗ A∨ 1) ∼ = Hom(A1 , A2 ). The symplectic counterpart Φt∗y therefore maps an indecomposable element of FKE τ with y = y1 τ + y2 as follows ¡ ¢ L = pr∗ (x(t), y(t)), α, M _
² ¡ ¢ pr∗ (x(t) + y1 , y(t) + (n − 1)y1 ), α, exp(−2πiy2 )M .
The map on the underlying Lagrangian is the translation Λ 7→ Λ + (y1 , (n − 1)y1 ). This is a one to one map, and since the local system are affected as well this has to be a composite of a map that only effects the underlying Lagrangian and a map that only twists the local system. We concentrate on the translation (i.e. assume y2 = 0) and suppose r = 1. Consider the following Lagrangians: (t, 2t) 7→ (t + y1 , t + y1 ) and
(t, 3t) 7→ (t + y1 , t + 2y1 ).
Note that the origin is mapped to (0, 0) 7→ (y1 , y1 ) and
(0, 0) 7→ (y1 , 2y1 ).
if this transformation comes from a morphism of the whole manifold, then the origin has one welldefined image. From (y1 , y1 ) ≡ (y1 , 2y1 ) follows y1 ∈ Z, and therefore y1 ≡ 0. So there is no symplectic counterpart of the functor t∗y1 τ +y2 if y1 6= 0. The same is true for the pushforward, since ty∗ = t∗−y . If y1 = 0, then what is left is a twist of all local systems with the factor exp(−2πiy2 ). This looks like a good morphisms, but it does not have a geometric interpretation in the sense that it is induced by a global geometric construction The natural construction on the symplectic side that only effects the local systems is the tensoring with the restriction of a globally given local system on E τ . This results in tensoring the monodromy with a morphism that depends on the slope of the underlying Lagrangian, though. We study this functor in more detail in chapter 9.
4.3
Tensoring with a line bundle
⊗n . The functor Let L be a line bundle on Eτ . Then L = L(t∗y ϕ0 ϕ−1 0 ) ⊗ L(ϕ0 ) that tensors each object with L is the composition of the translationfunctor t∗y
4.3 Tensoring with a line bundle
39
with n−times the functor that tensors with L(ϕ0 ). We have already treated the translation functor, so that it remains to investigate the functor · ⊗ L(ϕ0 ). Consider a special vector bundle on Eτ with x = aτ + b: L(t∗x ϕ0 ϕn−1 ) ⊗ F (V, exp N ) 0
Φτ
¡
/ (a + t, a(n − 1) + nt), α, (V, exp N )
¢
² ¢ Φτ / ¡ L(t∗x ϕ0 ϕn0 ) ⊗ F (V, exp N ) (a + t, an + (n + 1)t), α, (V, exp N ) ¡ ¢ In this case we see that Φ · ⊗ L(ϕ0 ) comes from the map (x, y) 7→ (x, x + y), which is a simple Dehn-twist of the torus. On morphisms the functor ⊗L(ϕ0 ) acts as the identity, which is also the action of the Dehn-twist. Also for general vectorbundles and their symplectic counterparts we get a commutative diagram: ¡ ¢ πr∗ L(t∗x ψ0 ψ0n−1 ) ⊗ F (V, exp N )
¡¡ ¢ ¢ / pr∗ a + t, a(n − 1) , α, M
Φτ
¡ ¢ (ra + t, a(n − 1) + nr t), α ˜ , pr∗ M
⊗L(ϕ0 )
² ¡ ¢ πr∗ L(t∗x ψ0 ψ0n−1 ) ⊗ F (V, exp N ) ⊗ L(ϕ0 )
¡
²
Dehn-twist
˜ pr∗ M (ra + t, a(n − 1) + nr t + ra + t), β,
¢
∼
²
¢ ¡ (n+r)−1 ) ⊗ F (V, exp N ) πr∗ L(t∗x ψ0 ψ0
Φτ
¡ ¢ / pr∗ (a + t, a(n + r − 1) + (n + r)t), β, M .
Tensoring a torsion sheaf S(x, V, N ) with L yields the torsion sheaf S(x, V, N ) again. Similarly for the symplectic counterpart: The underlying Lagrangian is (a, t) and this maps to (a, a + t) ∼ (a, t). This is a one to one map, and therefore the local system is as well not affected. So far we have proven that the tensoring with a line bundle and the Dehn-twist correspond on the level of objects. Now we have to prove the same for morphisms. On torsion sheaves it was the identity already on the level of objects and for F1 , F2 two vectorbundles we get: Hom(F1 , F2 ) ²
∼ =
H 0 (F2 ⊗ F1∨ )
/ Hom(F1 ⊗ L, F2 ⊗ L) ∼ =
² / H 0 (F f2 ⊗ F ∨ ⊗ L ⊗ L∨ ) 1
∼ =
/ H 0 (F2 ⊗ F ∨ ). 1
In both cases, for torsion sheaves and for vectorbundles, the functor operates as the identity. The Dehn-twist operates on the symplectic side also as the identity: Let L1 , L2 be symplectic counterparts of special vectorbundles. Then the morphism space Hom(L1 , L2 ) is generated by the intersection points ek ∈ Λ1 ∩ Λ2 , explicitly given by formula (2.4.1) as ³a − a + k n a − n a + n k´ 2 1 1 2 2 1 1 ek = , with 0 ≤ k < n2 − n1 . n2 − n1 n2 − n1
¡ ¢ Automorphisms of τ, Db (Eτ ) in pDb ell
40
These points are mapped under the Dehn-twist to ³a − a + k n a − n a + n k a − a + k´ 1 2 2 1 1 2 1 2 1 ek 7→ dk = , + n2 − n1 n2 − n1 n2 − n1 ³ a2 − a1 + k (n1 + 1)a2 − (n2 + 1)a1 + (n1 + 1)k ´ = , , (n2 + 1) − (n1 + 1) (n2 + 1) − (n1 + 1) which are the intersection points of the images of Λ1 , Λ2 under the Dehn-twist. Therefore also Mi |ek 7→ Mi |dk , and the Dehn-twist operates on the morphism spaces as the identity. We get by similar considerations that the Dehn-twist operates trivial on the morphisms spaces in general - if the objects are counterparts of general vectorbundles and torsion sheaves. Remark 4.1. We have here confirmed the remark of Polishchuk and Zaslow in [63], where they already stated without proof that tensoring with L(ϕ0 ) on the algebraic side corresponds to a simple Dehn-twist of the torus on the symplectic side.
4.4
Dualizing
Another natural functor is the contravariant functor that sends each coherent sheaf F to its dual F ∨ = RHom(F, OEτ ). For vector bundles the derived dual is the usual dual. Note that L(t∗x ϕ0 ϕn−1 )∨ ∼ = L(t∗−x ϕ0 ϕ0−n−1 ) 0 and further
¡ ¢ F (V, exp N )∨ = F V ∨ , exp(−N ∨ ) .
Finally note that ¡ ¢∨ ¡ ¢ πr∗ L(ϕ) ⊗ F (V, exp N ) = πr∗ (L(ϕ) ⊗ F (V, exp N ))∨ . This yields ¡ ¢ ¡ ¢ ) ⊗ F (V, exp N ) 7→ πr∗ L(t∗−x ϕ0 ϕ0−n−1 ) ⊗ F (V ∨ , exp −N ∨ ) . πr∗ L(t∗x ϕ0 ϕn−1 0 For vector bundles this functor yields the identity on morphisms, under the isomorphism: ¡ ¢ Hom(F2∨ , F1∨ ) ∼ = H 0 F1∨ ⊗ (F2∨ )∨ ∼ = Hom(F1 , F2 ). Applying twice the mirror functor this gives us with the usual notation ¡ ¢Â πr∗ L(ϕ) ⊗ F (V, exp N )
Φτ
/ pr∗ (a + t, a(n − 1) + nt), α, V, exp(−2πib + N )
² ¡ ¢Â πr∗ L(ϕ−1 ) ⊗ F (V ∨ , exp −N )
Φτ
/ pr∗ (−a + t, −a(−n − 1) − nt), −α, V ∨ , exp(2πib − N ∨ )
Note that
¡
¡
¡
¢¢
¡
¡ ¢ ¡ ¢ −a + t, −a(−n − 1) − nt ∼ a + t, −a(n − 1) − nt ,
¢¢
4.5 Involution
41
which is the image of Λ under the map rx : (x, y) 7→ (x, −y). This ¡ is not a symplectic ¢ τ , and it also fails to map the local system V, exp(−2πib+N ) to automorphism of E ¡ ∨ ¢ V , exp(2πib−N ∨ ) . The data of M = (W, M ) does not determine the local system completely, because the orientation of Λ is needed; we always use the convention that the orientation of Λ is given by exp(2πiβ) with β ∈ (− 21 , 21 ] such that α−β ∈ Z. Bearing this in mind, we see that rx∗ (V, M ) = (V, M ).
4.5
Involution
The isomorphism of Eτ sending z to −z induces pullback or pushforward a ¡ ∗ vian−1 ¢ b (E ), sending π natural equivalence of D L(t ϕ ϕ ) ⊗ F (V, exp N ) to the τ r∗ x 0 0 ¡ ¢ bundle πr∗ L(t∗−x ϕ0 ϕn−1 ) ⊗ F (V, exp −N ) and S(x, V, N ) 7→ S(−x, V, N ). To see 0 this, first note that πr commutes with the involution. A calculation on the fibres shows that a vectorbundle F (V, mτ ) with the multiplier mτ is sent to the bunn−1 −1 ∗ dle F (V, m−1 ) (−z − τ ) = τ (−z − τ )). An easy calculation shows that (tx ϕ0 ϕ0 (t∗−x ϕ0 ϕn−1 )(z). For the image of F (V, exp N ) this gives us the multiplier exp(−N ). 0 The description of the operation of the functor on morphisms is similarly simple again only the translation changes. For morphisms between line bundles we get: t∗x21 θ
£
k n2 −n1 , 0
¤¡ ¢ £ k ¤¡ ¢ (n2 − n1 )τ, (n2 − n1 )z 7→ t∗−x21 θ n2 −n , 0 (n2 − n1 )τ, (n2 − n1 )z , 1
1 with x21 = nx22 −x −n1 . This functor has a symplectic counterpart, induced by the symplectic involution (x, y) 7→ (−x, −y). As images under the mirror functor we get with M = −2πib + N and the usual definitions:
¡ ¢Â πr∗ L(t∗x ϕ0 ϕn−1 ) ⊗ F (V, exp N ) 0
_
² ¡ ¡ ¢¢ Â πr∗ L(t∗−x ϕ0 ϕn−1 ) ⊗ F V, exp(−N ) 0
³
/ pr∗ (a + t, a(n − 1) + nt), α, (V, exp M )
³
´
´
/ pr∗ (−a + t, −a(n − 1) + nt), α, (V, exp(−M ) .
And the functor by the symplectic involution gives us the same ³ ¡ ¢´ pr∗ (a + t, a(n − 1) + nt), α, M = V, exp(−2πib + N ) ² ³ ¡ ¢´ pr∗ (−a + t, −a(n − 1) + nt), α, M = V, exp(2πib − N ) .
Here we have to take care with the orientation of Λ. Though its slope stays the same, the orientation is reversed by the map. According to our convention, we therefore have to take the inverse monodromy for the local system on the image. A rigorous definition of the functor induced by symplectic maps is given in chapter 7. Since the involution commutes with pr we can assume r = 1 when calculating the action of the functor on the morphisms. For two symplectic counterparts of special vectorbundles the space of morphisms is
¡ ¢ Automorphisms of τ, Db (Eτ ) in pDb ell
42 generated by
L Ã 1 ∩ L2 = (a1 + t, a1 (n1 − a) + n1 t) ∩ (a2 + t, a2 (n2 − a) + n2 t) ´ o n ³ −a1 n1 a2 −n2 a1 +n1 k , , = ek := na22 −n n2 −n1 1 And the morphism space of the image in generated by (−a1 + t, −a1 (n1 − 1) + n1 t) ∩ (−a2 + t, −a2 (n2 − 1) + n2 t) n ³ ´ o 2 +a1 −n1 a2 +n2 a1 +n1 k = dk := −a , , n2 −n1 n2 −n1 and the functor maps simply ek 7→ dk . In combination with the mirror functor and the algebraic involution we get the commutative diagram. fk ⊗ T Â _ ² gk ⊗ T Â
Φτ
Φτ
/ C · T · ek _ ² / C · T · dk
with ´ ³ ³ ¡ ¢´ 2 ∨ a2 −a1 b2 −b1 1) ∨ ∨ ) exp N ⊗ 1 C = exp −πi( (an22−a − 1 ⊗ N − 2πi 1 V V1 V2 ⊗V1 n2 −n1 2 1 −n1 n2 −n1 2 ³ ´ ³ ¡ ¢´ (−a2 +a1 )2 ∨ −a1 b −b1 ∨ + 1V ⊗ N ∨ 2 = exp −πi( n2 −n1 ) exp na22 −n (−1) −N ⊗ 1 . + 2πi 1 V V ⊗V 2 1 2 n −n 2 1 1 1 2 1
With similar considerations we check that functors induced by the symplectic and algebraic involution also correspond for general vectorbundles and torsion sheaves which proves that the symplectic involution is mirror to the algebraic involution.
Chapter 5
Isogenies of Elliptic Curves Now we investigate the functors induced by a morphism of elliptic curves f : E1 − → E2 . Given f there exist two lattices (which we fix throughout this chapter) Λ1 , Λ2 in C, with Λ1 = Zrv + Zsw and Λ2 = Zv + Zw such that Ei = C/Λi and the following diagram commutes f
E1 ∼ =
/ E2 ²
²
∼ =
/ C/(Zv + Zw),
C/(Zrv + Zsw)
where the lower map is given by the inclusion of lattices. We also denote this map by f . Without loss of generality we can assume Im(v/w) > 0. Remark 5.1. If E1 and E2 are isomorphic, under the isomorphisms we used here, only the identity is left. The automorphisms of an elliptic curve therefore have to be treated separately; we do this in chapter 6. We can now decompose the morphism as shown in the following diagram: z z7→ sw
C/(Zrv + Zsw)
C/(Zv + Zsw) &
²
z z7→ sw
²
/ C/(Z v + Z) π sw γs
fs
C/(Zv + Zw)
(5.0.1)
πr
fr
²
f
/ C/(Z rv + Z) sw
z z7→ w
²
{ / C/(Z v + Z) w
fr , fs , πr are inclusions of the respective lattices, and γs is the following map: v Define τ = sw then γs : Eτ − → Esτ , z 7→ sz. In the opposite direction we have a different map that we already know: πs : Esτ − → Eτ , but this map does not fit into this diagram. Neither does it commute with the maps appearing here nor is it a form of inverse of γs or the induced functors adjoint.
44
Isogenies of Elliptic Curves
Remark 5.2. In chapter 2 we chose to trivialize all our vector bundles over the cylinder, which is given by C/Z and only keep a nontrivial multiplier over the other generator of the given lattice. Instead, we could have chosen to trivialize all our vector bundles over the cylinder, which is given by C/Zτ and have the nontrivial multiplier over the generator 1 of our lattice. For each isomorphism class of vector bundles we would then have a representative of the form γs∗ (A), where A is a vector bundle over E τs and of the form F (V, 1, f (z) exp N ). And like in our description we could then use a special f0 , such that we could describe all line bundles as products of F (C, 1, f0 ) and appropriate pullbacks. Then we could again choose a standard basis for the global sections of these bundles and build a similar theory like we have done before. All maps in diagram 5.0.1 induce functors on the categories of coherent sheaves, the direct image and inverse image functors; and the horizontal arrows induce even equivalences because they are isomorphisms: ¡ ¢ Db C/(Zrv + Zsw) ²
f∗
∼
/ D b (E ) rτ πr∗
fr∗
¡ ¢ Db C/(Zv + Zsw)
²
∼
/ D b (E ) τ
π∗
γs∗
fs∗
² ¡ & ¢ Db C/(Zv + Zw)
²
{ / D b (E ) sτ
∼
The map f is finite and therefore the direct image and inverse image functors are exact, see Grauert Remmert [32, Exactness Lemma], and therefore we can interpret them as functors on the derived categories. In the future we will call these functors pushforward and pullback. To investigate the pushforward and pullback with respect to these morphisms, it suffices to treat the morphisms πr and γs , separately, because also on the Fukaya side, we can decompose the counterpart of the functor, as is pictured in the following diagram ¡ ¢ Db C/(Zrv + Zsw)
f∗
Db
²
∼
/ D b (E ) Φrτ / FK(E rτ ) rτ E πr∗ π∗
fr∗
¡ ¢ C/(Zv + Zsw)
∼
²
/ D b (E ) τ γs∗
fs∗
² ¡ & ¢ Db C/(Zv + Zw)
∼
² ¢
pr∗
Φτ
/ Db (E ) Φsτ / FK(E sτ ) sτ
We want to construct P such that P = Φsτ ◦ π∗ ◦ Φ−1 rτ and G such that G = Φsτ ◦ γs∗ ◦ Φ−1 τ .
:
,  / FK(E τ ) P  µ  ¥ G ² { y ²
5.1 The functors πr∗ and πr∗
45
Then P = G ◦ pr∗ . After that we have to ask, whether there are symplectic maps p : E rτ − → E sτ and g : E τ − → E sτ such that G = g∗ and P = p∗ . For the pullback we get a similar diagram with all arrows reversed and lower star changed to upper star.
5.1
The functors πr∗ and πr∗
We already know that the counterpart of πr∗ is the functor pr∗ as defined in section 2.3. Because the pushforwards and pullbacks are adjoint, therefore also the pullbacks correspond to each other. We only have a brief look at how objects are mapped: Consider the bundle L(t∗x ϕ0 ϕn−1 ) ⊗ F (V, exp N ) with x = aτ + b: Then, because 0 r X
n( 12 + 12 τ −
x n
+ kτ ) = ( 12 + 12 rτ − rx) + (nr − 1)( 12 + 12 rτ )
k=1
we calculate for the divisor according to proposition C.3 πr∗ (n[ 12 + 12 τ − nx ]) =
r X
n[ 21 + 12 τ − x + kτ ]
k=1
∼ [ 21 + 12 rτ − rx] + (nr − 1)[ 21 + 21 rτ ] This yields for the corresponding line bundle πr∗ L(t∗x ϕ0 ϕ0n−1 ) = L(t∗rx ψ0 ψ0rn−1 ) and we get together with the mirror functors L(t∗x ϕ0 ϕn−1 ) ⊗ F (V, exp N ) 0
Â
Φτ
_ ²
∗ πr
¡ ¢Â L(t∗rx ψ0 ψ0nr−1 ) ⊗ F V, exp(rN )
Φτ
¡
¢
/ (a + t, a(n − 1) + nt), α, (V, exp(−2πib + N )) Â Â Â ² ¡
¢
/ (ra + t, ra(rn − 1) + rnt), β, (V, exp(−2πirb + rN ))
Define ¡ ¢ ¡ ¢ (a + t, a(n − 1) + nt), α, (V, exp(−2πib + N )) =: Λ, α, M =: L ¡ ¢ ¡ ¢ (ra + t, ra(rn − 1) + rnt), β, (V, exp(−2πirb + rN )) =: Λ0 , α0 , M0 =: L0 So we have to show that the lower element L0 in FK(E rτ ) is the pullback of the upper along pr . Potentially, p−1 r Λ might have several components. First we note that all of these components are contained in the set p−1 r Λ=
n¡ a r
+
k r
o ¢ + t, a(n − 1) + nrt , k = 1, . . . , r .
A simple calculation shows that this set contains only one Lagrangian. ¡ ¢ Denote by γΛ a lift of the positive generator of π1 (Λ) with starting point a, a(n−1) ¢ ¡ and analogously γp−1 is the lift of the positive generator of π1 p−1 r (Λ) with r (Λ)
46
Isogenies of Elliptic Curves
starting point
¡a
n , a(n
¢ − 1) . Then we calculate γΛ : [0, 1] − → Eτ ¡ ¢ t 7→ a + t, a(n − 1) + nt
p−1 → E rτ r (γΛ ) : [0, 1] − ¡ ¢ t 7→ ar + rt , a(n − 1) + nt γp−1 : [0, r] − → E rτ r (Λ) ¡ ¢ t 7→ ar + rt , a(n − 1) + nt
we see that we have run r-times along p−1 once. r (Λ) when we have run along γp−1 r (Λ) ∗ Therefore we get for the pullback of the local system pr M = (V, exp(−2πib + N )r ), which is exactly what we wanted to show. For more general vector bundles this works as well, because we apply the mirror functor anyway only to the special bundle. For torsion sheaves we used in the previous chapter the identity ¡ ¢ ¡ ¢ Φrτ πr∗ S(x, V, N ) = p∗r Φτ (S(s, V, N )) . And in this case the inverse image πr∗ S(x, V, N ) is given by πr∗ S(x, V, N )
=
r M
S(x + kτ, V, N ).
k=1
This yields (with x = aτ + b) Φrτ πr∗ S(x, V, N ) =
r M
Φrτ S(x + kτ, V, N )
k=1 r ³ M ¡ ¢´ a+k 1 = (− r , t), 2 , V, exp(2πib + N ) k=1
On the other hand ³ ¡ ³ ¢´ ¡ ¢´ p∗r Φτ S(x, V, N ) = p∗r (−a, t), 12 , V, exp(2πib + N ) =
r ³ M ¡ ¢´ 1 , t), , V, exp(2πib + N ) . (− a+k r 2 k=1
5.2
The functors γs∗ and γs∗
The map γ induces by pullback and pushforward functors between the respective derived categories. The aim of this section is to describe these functors explicitly. Only in the next section we will care about whether there is a symplectic counterpart for them. Lemma 5.3. The pushforward and pullback of γ commute with the pushforward of πr :
5.2 The functors γs∗ and γs∗
47
1. γs∗ πr∗ (·) = πr∗ γs∗ (·) 2. γs∗ πr∗ (·) = πr∗ γs∗ (·) Proof. Notice that the πr on the right hand side of the equations are different from the ones on the left hand side. To be precise, we should have mentioned the respective categories on which the functors are defined. The following diagram shows the maps between the elliptic curves for the first equation: < Eτ FF FF γ xx x FF xx FF x xx " πr
Erτ F
FF FF γ FFF "
Ersτ
Esτ x< x xx xx πr x x
The lemma follows from the fact that the underlying morphisms of elliptic curves commute, which can be checked by a simple calculation. For the second equation we have the diagram: .
Erτ C
x xx xxγ x x| x
CC CC πr CCC !
FF FF πr FFF "
{{ {{ { { γ }{{
ErsτF
Esτ
Eτ
Again, the claim follows from the commutativity of this diagram and flat base change. Lemma 5.4. Define the line bundles L+− = Fτ (C, 1, −1) L−− = Fτ (C, −1, −1). Both have degree zero and therefore are C ∞ -equivalent to the trivial bundle. As holomorphic bundles they are isomorphic to ∨ ∗ ∼ L+− ∼ = L(t∗τ ϕ0 ϕ−1 0 ) = L(ϕ0 ) ⊗ L(t τ ϕ0 ) 2
2
∨ ∗ ∼ L−− ∼ = L(t∗τ +1 ϕ0 ϕ−1 0 ) = L(ϕ0 ) ⊗ L(t τ +1 ϕ0 ), 2
2
where as before ϕ0 (z) = exp(−πiτ − 2πiz). Proof. Let L denote either of the two bundles. L⊗L is trivial; therefore deg(L⊗L) = 0 = 2 deg(L). Also their Chern-class is zero, see remark C.6, which classifies C ∞ line bundles completely. It follows that there is a C ∞ -isomorphism to the trivial
48
Isogenies of Elliptic Curves
bundle. Explicit holomorphic isomorphisms are given by: L+− − → L(t∗− τ ϕ0 · ϕ−1 0 ), [z, v] 7→ [z, exp(−πiz)v] 2
−−
L
− → L(t τ +1 ϕ0 · ϕ−1 0 ), [z, v] 7→ [z, exp(−πiz)v] ∗
2
Lemma 5.5. Let ϕ = t∗x ϕ0 ϕn−1 . Then 0 ¡ ¢ γs∗ Lτ (ϕ) = Fsτ (Cs , A, B) =: E, where and¢B are given with respect to a basis {e1 , . . . , es } of Cs as follows. B(z) = ¡ Az+k diag ϕ( s ) and A is the (constant for z) morphism which permutes the basis vectors cyclicly. More explicitly: z ϕ( s ) 0 ϕ( z+1 s ) B(z) = , . . . z+s−1 0 ϕ( s ) 0 0 ... 0 1 1 0 0 . . . 0 0 1 0 . . . 0 A = . .. . .. .. . . . .. 0 ... Further
¡ det E = L t∗
sx+
0
1
0
n(s−1) (s−1) + 2 sτ 2
¢ ψ0 ψ0n−1 .
¡ ¢ ¡ ¢ Proof. To see that γs∗ Lτ (ϕ) = Fsτ Cs , A, B(z) =: E, just do the computation on fibres. Denote by ψ0 the function ψ0 (z) = exp(−πi(sτ ) − 2πiz) = ϕ0 (sτ, z). For the determinant we first calculate det B(z) = =
s−1 Y k=0 s−1 Y
ϕ( z+k s ) exp(−2πix) exp(−πinτ − 2πin z+k s )
k=0
= exp(−2πisx − 2πi n(s−1) ) exp(−πinsτ − 2πinz) 2 = t∗
sx+
n(s−1) 2
and det A = (−1)s−1
ψ0 ψ0n−1 (z).
5.2 The functors γs∗ and γs∗
49
And so we find ¡ ¢ ¡ ¢ det E = F C, det B(z), det A = F C, det B(z), 1 ⊗ F (C, 1, det A) ¡ ¢⊗(s−1) = L(t∗ n(s−1) ψ0 ψ0n−1 ) ⊗ L+− sx+
2
=
L(t∗ n(s−1) ψ0 ψ0n−1 ) sx+ 2
⊗ L(t∗1 sτ ψ0 ψ0−1 )⊗(s−1)
=
L(t∗ n(s−1) ψ0 ψ0n−1 ) sx+ 2
⊗ L(t (s−1) ψ0 ψ0−1 )
=
L(t∗ n(s−1) (s−1) ψ0 ψ0n−1 ) sx+ 2 + 2 sτ
2
∗
2
sτ
Though we now know the image of L(ϕ) under γs∗ , it is obviously not described in our preferred notation. This representative is given by the following proposition. Proposition 5.6. Let ϕ = t∗x ϕ0 ϕn−1 like above with n 6= 0. The preferred repre0 sentative of the isomorphism class of γs∗ L(ϕ) can then be described as follows. There exists an N ∈ End(Ch ) with dim ker N = 1, h = gcd(s, n), such that for hr = s, hd = n we have ¡ ¢ ¡ ¢ γs∗ L(ϕ) ∼ = πr∗ L(ξ) ⊗ F (Ch , exp N ) with ξ = t∗y ξ0 · ξ0d−1 and ξ0 (z) = ϕ0 (rsτ, z) = exp(−πi(rs)τ − 2πiz) and y = rx −
n(s − 1) + (r − 1) (s − 1) + n(r − 1) − sτ 2h 2h
If n = 0 we have with ϕ = t∗x ϕ0 ϕ−1 0 ¡ ∗ ¡ ¢ ¢ −1 ∼ γs∗ L(t∗x ϕ0 ϕ−1 0 ) =πs∗ L(tsx+ s−1 sτ ξ0 ξ0 ) .
(5.2.1)
2
Proof. First note that F (Cs , B(z), A) is indecomposable (there is no subspace of Cs , such that this subspace is invariant under the operation of A). For n 6= 0 we use Lemma 5.5 and Atiyah [3, Lemma 24] to see that there exist y ∈ Ersτ and N ∈ End(V ) with dim ker N = 1 such that ¡ ¢ ¡ ¢ γs∗ L(ϕ) ∼ = πr∗ L(t∗y ξ0 ξ0d−1 ) ⊗ F (Ch , exp N ) , Now we have to determine y such that ³ ¡ ¡ ¡ ¢¢ ¢´ det γs∗ L(ϕ) = det πr∗ L(t∗y ξ0 ξ0d−1 ) ⊗ Fsτ (Ch , exp N ) . ¡ ¡ ¢¢ We have already determined det γs∗ L(ϕ) in lemma 5.5 to be L(ψ), where ψ = t∗α ψ0 ψ0n−1 with: α = sx −
n(s − 1) s − 1 − sτ. 2 2
50
Isogenies of Elliptic Curves
So now we find: ³ ¡ ¡ ¢ ¢´ det πr∗ Lrsτ (ξ) ⊗ Fsτ (Ch , exp N ) = F C, det(πr∗ (ξ exp N ) ¡ ¢ = F C, (−1)r−1 ξ h . and ¡ ¢h (−1)r−1 ξ(z)h = (−1)r−1 t∗y ξ0 ξ0d−1 (z) ¡ ¢ = exp −πi(r − 1) exp(−2πihy) exp(−πinrsτ − 2πinz) ¢ ¡ (r−1) = exp −2πi(hy + r−1 2 + 2 nsτ ) exp(−πinsτ − 2πinz) We want this to equal = exp(−2πiα) exp(−πinsτ − 2πinz) This yields for y: α = hy +
r−1 2
+
(r − 1)n sτ 2
r − 1 (r−1)n − 2 sτ 2 n(s − 1) + (r − 1) (s − 1) + n(r − 1) − sτ = sx − 2 2 n(s − 1) + (r − 1) (s − 1) + n(r − 1) ⇔ y = rx − − sτ 2h 2h
⇔ hy = α −
For n = 0 we know that γs∗ (L(t∗x ϕ0 ϕ−1 0 )) is isomorphic to a bundle of the form πs∗ (L(t∗y ξ0 ξ0−1 )). We then determine y by comparing the determinants, which yields y = α, which is in this case sx − s−1 2 sτ . ¡ ¢ Proposition 5.7. Let ϕ0 and ψ0 be defined as above. Then γs∗ L¡sτ (ψ)¢ = Lτ (ϕ) with ψ¡ = t∗x¢ψ0 · ψ0n−1 and ϕ = t∗x ϕ0 · ϕns−1 . So we have rank γs∗ L(ψ) = 1 and 0 ∗ deg γs L(ψ) = s · deg L(ψ). Proof. The proof follows from a simple calculation on the fibres. Just note that ψ(sz) =t∗x ψ0 ψ0n−1 (sz) = exp(−2πix) exp(−πinsτ − 2πinsz) =t∗x ϕ0 ϕns−1 (z) 0 =ϕ(z).
Corollary 5.8. We can calculate the image of a special vector bundle und the pushforward of γ according to the following formula: ¡ ¢ ¡ ¢ 1. γs∗ L(ϕ) ⊗ F (V, exp N ) = γs∗ L(ϕ) ⊗ F (V, exp N ) ¡ ¢ ¡ ¢ 2. γs∗ L(ψ) ⊗ F (V, exp N ) = γs∗ L(ψ) ⊗ F (V, exp N )
5.2 The functors γs∗ and γs∗
51
Proof. We use the same notation as in lemma 5.5. By a direct computation on the fibres, we see that ¡ ¢ ¡ ¢ γs∗ L(ϕ) ⊗ F (V, exp N ) = γs∗ F (V, ϕ exp N ) = F (V s , C(z, N ), A). where
ϕ( zs ) exp N v1 v1 .. .. C(z, N ) . = .
.
) exp N vs ϕ( z+s−1 s
vs
On the other hand a similar calculation yields: ¡ ¢ γs∗ L(ϕ) ⊗ F (V, exp N ) = F (Cs , B(z), A0 ) ⊗ F (V, exp N ) = F (Cs ⊗ V, B(z) ⊗ exp N, A0 ⊗ 1V ) ∼ = F (V s , C(z, N ), A). With the notation of proposition 5.7 we see similarly ¡ ¢ ¡ ¢ γs∗ L(ϕ) ⊗ F (V, exp N ) = γs∗ F (V, ϕ exp N ) = F (V, ϕ exp N ) ¡ ¢ = γs∗ L(ψ) ⊗ F (V, exp N ).
This means we can now determine the image of an arbitrary vector bundle for the functors γs∗ and γs∗ . For torsion sheaves, we get Proposition 5.9. For torsion sheaves Ssτ (x, V, N ) respectively Sτ (x, V, N ) the images under pullback, respectively pushforward are: 1. γs∗ Ssτ (x, V, N ) = S(sx, V, N ), L 2. γs∗ Sτ (x, V, N ) = γ(y)=x S(y, V, N ) Proof. All fibres except at the image resp. inverse image of the support are zero. So we get this result. Remark 5.10. The description of how γ maps morphisms is much easier than the description of the action on objects. We use a flat base change, similar to the one used in the construction of the morphisms space of preferred vector bundles before. Consider the following diagram, where γ˜i is the projection on the first and the second component, respectively. Eτ × Z/sZ ∼ = i
²
γ ˜2
Eτ ×Esτ Eτ ²
γs
γ ˜1
Eτ
/ Eτ
γs
²
/ Esτ
52
Isogenies of Elliptic Curves
Define the morphisms γ˜i,µ = γ˜i |Eτ ×{µ} . Then we can use flat base change to get the following identities (here we only claim existence of ζi,ν ): ³ ´ Hom L(ϕ1 ) ⊗ F (V1 , exp N1 ), L(ϕ2 ) ⊗ F (V2 , exp N2 )
(5.2.2)
γs∗
² ³ ¡ ¢ ¡ ¢´ Hom γs∗ L(ϕ1 ) ⊗ F (V1 , exp N1 ) , γs∗ L(ϕ2 ) ⊗ F (V2 , exp N2 )
Ls−1 ν=0
³ ¡ ¢ ∗ ¡ ¢´ ∗ Hom γ˜s,ν L(ϕ1 ) ⊗ F (V1 , exp N1 ) , γ˜2,ν L(ϕ2 ) ⊗ F (V2 , exp N2 )
Ls−1 ν=0
¡ ¢ Hom L(ζ1,ν ) ⊗ F (V, exp N1 ), L(ζ2,ν ) ⊗ F (V2 , exp N2 ) O ν ζ,N
Ls−1 ν=0
¡ ¢ −1 H 0 L(ζ2,ν ζ1,ν ) ⊗ (V2 ⊗ V1∨ )
ζi can be calculated explicitly, once we have decided, which isomorphism i we want to choose. In section 6.2 we will need a specific choice of an isomorphism, which is given in lemma 6.11.¡ Following this lemma, we¢ calculate then the more general morphism spaces Hom γs1 ∗ (L1 ⊗ F1 ), γs2 (L2 ⊗ F2 ) . Independently from this choice we see that s−1 M γs∗ : fk ⊗ T 7→ gk,ν ⊗ T, ν=0
¡ ¢ −1 where {gk,ν } is a our standard basis of H 0 L(ζ2,ν ζ1,ν ) .
5.3
The symplectic counterpart of γ
To extend the equivalence Φτ to a functor Φ : pDb ell − → pFKell we need to construct a symplectic counterpart of γs∗ and γs∗ and check, whether it is a morphism in pFKell. However, the symplectic counterparts have no geometric interpretation and are therefore not contained in pFKell. Theorem 5.11. The symplectic counterparts of the functors γs∗ and γs∗ have no geometric interpretation. The proof is elementary - in chapter 7 we investigate all functors on the Fukaya categories induced by morphisms of the underlying manifolds and we will see that none of them corresponds to the functors described here. In chapter 9 we will describe more functors between the Fukaya categories but nevertheless we will not find the counterpart we need. This means we found a new functor between the Fukaya categories by the use of the mirror functor. It maps objects corresponding to line bundles as follows. Let L be
5.3 The symplectic counterpart of γ
53
the counterpart of L(t∗x ϕ0 ϕn−1 ) with x = aτ + b, i.e. 0 ³ ¡ ¢´ L = (a + t, a(n − 1) + nt), α, C, exp(−2πib) . ¡ ¢ Denote by L0 the counterpart of the vectorbundle γs∗ L(t∗x ϕ0 ϕ0n−1 ) . Then we get according to our above calculations and with the above notation h = gcd(s, n), hr = s, hd = n : ¡ ¢ L0 = pr∗ Λ, α0 , M , with Λ=
¡a s
−
(s−1)+n(r−1) 2s
(s−1)+n(r−1) )(d 2s ¢ n(s−1)+(r−1) ) + N) . 2h
+ t, ( as −
¡ M = Ch , exp(−2πi(b −
¢ − 1) + dt ,
So we see that a corresponding functor has to effect as well the underlying Lagrangian as the local system. In chapter 9 we will have functors that come from an underlying morphism combined with tensoring by a vectorbundle. But one can already see here that the action on the underlying Lagrangian does not come from a morphism of the complexified tori. For the corresponding map between the morphisms we see that (5.2.2) yields an identification of the intersection points of theLoriginal objects and their images under G. Then an element T · ek is mapped to s−1 ν=0 C(k, ν)T · ek,ν , where the factors C(k, ν) are given by conjugation with the respective correction terms of the mirror functor, this can be calculated after the choice of i in 6.2.
54
Isogenies of Elliptic Curves
Chapter 6
Automorphisms of the Elliptic Curve In the previous chapter we investigated isogenies between different elliptic curves and realized that the natural construction between the respective derived categories has no symplectic counterpart in pFKell. For our construction, as well as for the definition of the mirror functor for an elliptic curve E itself (see chapter 2), we had to make a choice of a lattice Λ and an oriented basis {v, w}, such that Λ = Zv + Zw and E ∼ = C/Λ. To investigate further how exactly morphisms operate on the derived categories, we will take a closer look at how automorphisms of the elliptic curve induce functors. With other words, in this chapter we are going to study, how the mirror functor depends the choices in volved in its construction. Given an elliptic curve, there is no natural choice of a lattice and its basis. We can always choose on one generator of the lattice being 1, but still there is an SL2 (Z) action on the upper complex half plane, yielding different bases and maybe different lattices for the same elliptic curve. The operation on the bases of lattices is given by µ ¶ aτ + b a b τ= c d cτ + d and is generated by the changes τ 7→ τ + 1 and τ 7→ − τ1 . We will compute the functors that correspond to these two actions at the algebraic side and then construct their symplectic counterparts on the Fukaya category. In 1 the symplectic category, E τ and E − τ are in general not isomorphic, so that on the symplectic side we are dealing with equivalences between categories of different complexified symplectic manifolds. We will prove that these equivalences are not natural on the symplectic side in the sense that they come from a morphism of the underlying manifolds - in general there is no morphism at all between the underlying manifolds.
6.1
The functors corresponding to τ 7→ τ + 1
The change of base τ 7→ τ + 1 of the lattice induces via direct image and inverse image functors, (auto)equivalences of the derived category of the elliptic curve. First
56
Automorphisms of the Elliptic Curve
we concentrate on the pushforward.
6.1.1
The pushforward and its symplectic counterpart
We will write: b∗ : Db (Eτ ) − → Db (Eτ +1 ), as if we had two different curves, because our description of the derived categories depends on the choices of basis for our morphism spaces, which is different for the two choices of the basis for the lattice. We choose also different representatives for the isomorphism classes of ¢vectorbundles. In Db (Eτ ) the preferred vector bundles ¡ are πr∗ L(ϕ) ⊗ F (V, exp N ) with ϕ = t∗x¡ϕ0 (rτ ) · ϕ0 (rτ )n−1 ; ¢in Db (Eτ +1 ) we want to write the vector bundles in the form πr∗ L(ζ) ⊗ F (V, exp N ) , where ζ = t∗y ϕ0 (r(τ + 1)) · ϕ0 (r(τ + 1))n−1 . The map b : C/(Zτ + Z) − → C/(Z(τ + 1) + Z), z 7→ z, is the identity on the covering C, but in the quotient we have z = aτ + b 7→ z = a(τ + 1) + (b − a). Proposition 6.1. The base change τ 7→ τ + 1 induces the functor b∗ : Db (Eτ ) −→ Db (Eτ +1 ). It can be described as follows. 1. An indecomposable vector bundle is given by πr∗ (L(ϕ) ⊗ F ), where L(ϕ) ⊗ F is a special vector bundles over Erτ with ϕ0 (z) defined as exp(−πirτ − 2πiz) = ϕ(rτ, z) and ϕ = t∗x ϕ0 ϕn−1 . Define further ζ0 (z) to 0 equal exp(−πir(τ + 1) + 2πiz) = ϕ(r(τ + 1), z). With this notation the base change is given by: πr∗
¡
Ob(Db Eτ ) ¢ L(t∗x ϕ0 ϕ0n−1 ) ⊗ F (V, exp N )
−→ Ob(Db Eτ +1 ) ¡ ¢ ζ0 ζ0n−1 ) ⊗ F (V, exp N ) , 7→ πr∗ L(t∗x+ rn 2
2. Recall that morphisms in the case ¡ ¢ Ai = πri L(ϕi ) ⊗ F (Vi , exp Ni ) can be viewed (see proposition 2.14) as a direct sum of morphisms between r1 r2 objects on Erτ , where r = gcd(r : 1 ,r2 ) ³ ¡ ¢ ¡ ¢´ Hom πr1 ∗ Lr1 τ (ϕ1 ) ⊗ Fr1 τ (V1 , exp N1 ) , πr2 ∗ Lr2 τ (ϕ2 ) ⊗ Fr2 τ (V2 , exp N2 ) gcd(r1 ,r2 )
∼ =
M
¡ r0 n −r0 n −1 ¢ H 0 (L t∗y(ν) ζ0 ζ0 2 1 1 2 ) ⊗ V2 ⊗ V1∨ ,
ν=1
where ζ0 (z) = ϕ0 (rτ, z) and y(ν) = r10 x2 − r20 x1 + n2 r10 ντ . Then ¡ ¢ k b∗ (fk,ν ⊗ T ) = exp πirk(1 − r0 n2 −r 0 n ) gk,ν ⊗ T 1 1
2
6.1 The functors corresponding to τ 7→ τ + 1
57
for each of the components of the direct sum, whith ¡ 0 ¢ 0 0 0 k fk,ν = t∗ y(ν) θ[ r0 n2 −r 0 n , 0] (r1 n2 − r2 n1 )rτ, (r1 n2 − r2 n1 )z 1 0 n −r 0 n r1 2 2 1
gk,ν = t∗
w(ν) 0 n −r 0 n r1 2 2 1
w(ν) = r10 (x2 +
1
2
¡ 0 ¢ 0 0 0 k θ[ r0 n2 −r 0 n , 0] (r1 n2 − r2 n1 )r(τ + 1), (r1 n2 − r2 n1 )z 1 1
r2 n2 2 )
2
− r20 (x1 +
r1 n1 2 )
+ n2 r10 ν(τ + 1)
3. for Torsion sheaves: Ob(Db Eτ ) −→ Ob(Db Eτ +1 ) S(x, V, N )
7→
S(x, V, N )
4. For two torsion sheaves with different support, the spaces of morphisms are zero anyway, 5. We still have to view the cases of Hom(A1 , A2 ), where either A1 or A2 are torsion sheaves. If A1 is a torsion sheaf, this is the trivial group. If A1 is not a torsion sheaf, but A2 is, then Hom(A1 , A2 ) ∼ = V2 ⊗ V1∨ ∼ = Hom(b∗ A1 , b∗ A2 ) ∨ and b∗ operates on V2 ⊗ V1 as the identity. Proof. We have to show that the described functor really is the pushforward of b. To this end, we proceed step by step: a. First consider only line bundles. In this case the assertion reads: Ob(Db Eτ ) − → Ob(Db Eτ +1 ) Lτ (t∗x ϕ0 · ϕn−1 ) 0
7→
Lτ +1 (t∗x+ n ζ0 · ζ0n−1 ) 2
The divisor corresponding to Lτ (ϕ) is 1+τ Dτ (ϕ) = ( 1+τ 2 − x) + (n − 1)( 2 )
and we calculate i h i ¡ ¢ h 1+(τ +1)−1 b Dτ (ϕ) = 1+(τ +1)−1 − x + (n − 1) 2 2 h i h i ∼ 1+(τ2+1) − (x + n2 ) + (n − 1) 1+(τ2+1) = Dτ +1 (ζ). (For the equivalence of the divisors see proposition C.3). We get the same result by the calculation ϕ(z) = ζ(z). b. Let Lτ (ϕ1 ), Lτ (ϕ2 ) be two line bundles on Eτ . Then we have to show: Mor(Db Eτ ) −→ Mor(Db Eτ +1 ) ¢ ¡ ¢ ¡ ¢ ¡ ¢ Hom L(ϕ1 ), L(ϕ2 ) = H 0 L(ϕ2 ϕ−1 −→ H 0 L(ζ2 ζ1−1 ) = Hom L(ζ1 ), L(ζ2 ) 1 ) ¡ ¢ k fk 7→ exp πik(1 − n2 −n ) gk , 1 ¡
58
Automorphisms of the Elliptic Curve where fk = t∗x2 −x1 θ n2 −n1
and gk = t∗y2 −y1 θ n2 −n1
£ £
¤¡
k n2 −n1 , 0
(n2 − n1 )τ, (n2 − n1 )z
¢
¡ ¢ (n2 − n1 )(τ + 1), (n2 − n1 )z .
k n2 −n1 , 0]
We know that ¡ ¢ ¡ ¢ Hom L(ϕ1 ), L(ϕ2 ) = H 0 L(ϕ2 ϕ−1 1 ) Define n = n2 − n1 , x = x2 − x¡1 , so that¢ ϕ2 ϕ−1 = t∗x ϕ0 ϕ0n−1 . Then the 1 −1 0 vectorspace of global sections H L(ϕ2 ϕ1 ) has the basis {fk = t∗x θ[ nk , 0](nτ, nz)|k = 0, . . . , n − 1} and it suffices to define the linear map ¡ ¢ ¡ ¢ Hom L(ϕ1 ), L(ϕ2 ) − → Hom b∗ L(ϕ1 ), b∗ L(ϕ2 ) on these basis vectors. The analogous basis of the target space is in the theorem denoted by gk : gk = t∗y θ[ nk , 0](n(τ + 1), nz), k = 0, . . . , n − 1 y = x2 − x1 +
n2 −n1 2
=x−
n 2
Before we start our calculation note that for all m, n ∈ Z the following is true: exp(πim2 n) = exp(πimn), because the value only depends on the parity of m and n. We calculate: gk = θ[ nk , 0](n(τ + 1), nz + x − n2 ) X ¡ ¢ = exp πi(m + nk )2 n(τ + 1) + 2πi(m + nk )(nz + x − n2 ) m∈Z
=
X
m∈Z
=
X
m∈Z
=
X
¡ ¢ ¡ ¢ exp πi(m + nk )2 nτ + 2πi(m + nk )(nz + x) · exp πi(m + nk )2 n − 2πi(m + nk ) n2 ¡ ¢ ¡ exp πi(m + nk )2 nτ + 2πi(m + nk )(nz + x) · exp πi(m2 n +
k2 n
¢ + 2mk − mn − k)
¡ ¢ ¡ ¢ exp πi(m + nk )2 nτ + 2πi(m + nk )(nz + x) · exp −πik(1 − nk )
m∈Z
¡ ¢ = exp −πik(1 − nk ) fk ,
which proves the assertion. c. Now consider a special vector bundle over Eτ . We have to show: Ob(Db Eτ ) − → Ob(Db Eτ +1 ) L(t∗x ϕ0 · ϕn−1 ) ⊗ F (V, exp N ) 0
7→
L(t∗x+ n ζ0 · ζ0n−1 ) ⊗ F (V, exp N ) 2
6.1 The functors corresponding to τ 7→ τ + 1
59
After the above calculations we know L(ϕ) = L(ζ). To see that Fτ (V, exp N, 1) = Fτ +1 (V, exp N, 1) we can calculate the effect of the base change on the multipliers eτ +1 = e1 ◦ eτ = 1 · exp N = exp N = eτ . ¡ ¢ d. We identify Hom L(ϕ1 ) ¡⊗ F (V1 , exp N1 ), L(ϕ2 ) ⊗ F (V2 , exp N2 ) according to ¢ ∨ proposition 2.9 with H 0 L(ϕ2 ϕ−1 1 ) ⊗ (V2 ⊗ V1 ) by the isomorphisms ν. The terms before the tensor already have been shown to be equal, the terms after the tensor are identical and correspond to one another by the identity map, since the two isomorphisms ν and b∗ ν are equal. e. Now we consider general vectorbundles on Eτ . Instead of working on Eτ or Eτ +1 we study the base change Erτ − → Er(τ +1) ; in analogy to the calculation in number 1, we have: ¡ ¢ ϕ(z) = exp −πirτ + 2πi(z + x) exp(−πi(n − 1)rτ + 2πi(n − 1)z) = exp(2πix) exp(−πinrτ + 2πinz) ¢ ¡ = exp 2πi(x − nr 2 ) exp(−πinr(τ + 1) + 2πinz) = ζ(z). f. For morphisms between special vectorbundles we have to show: Mor(Db Eτ ) − → Mor(Db Eτ +1 ) Hom(L(ϕ1 ) ⊗ F1 , L(ϕ2 ) ⊗ F2 ) Hom(L(ζ1 ) ⊗ F1 , L(ζ2 ) ⊗ F2 ) ¡ ¢ ¡ ¢ −1 0 ∨ = H L(ϕ2 ϕ1 ) ⊗ (V2 ⊗ V1 ) − → H 0 L(ζ2 ζ1−1 ) ⊗ (V2 ⊗ V1∗ ) = ¢ ¡ k fk ⊗ T 7→ exp πik(1 − n2 −n ) gk ⊗ T, 1 where Fi = F (Vi , exp Ni ). Denote by ϕ1 = t∗x1 ϕ0 (r1 τ )ϕ0 (r1 τ )n1 −1 ϕ2 = t∗x2 ϕ0 (r2 τ )ϕ0 (r2 τ )n2 −1 ¡ ¢ ¡ ¢n −1 ζ1 = t∗x1 + r1 n1 ζ0 r1 (τ + 1) ζ0 r1 (τ + 1) 1 2 ¡ ¢ ¡ ¢n −1 ∗ ζ2 = tx2 + r2 n2 ζ0 r2 (τ + 1) ζ0 r2 (τ + 1) 2 2
According to our calculations in e: ³ ¡ ´ ¢ Hom b∗ πr1∗ (L(ϕ1 ) ⊗ F (V1 , exp N1 )) , b∗ (πr2∗ (L(ϕ2 ) ⊗ F (V2 , exp N2 ))) = ³ ¡ ¢ ¡ ¢´ Hom πr1∗ L(ζ1 ) ⊗ F (V1 , exp N1 ) , πr2∗ L(ζ2 ) ⊗ F (V2 , exp N2 )
60
Automorphisms of the Elliptic Curve and so we have to define the morphism ³ ¡ ¢ ¡ ¢´ Hom πr1 ∗ Lr1 τ (ϕ1 ) ⊗ F (N1 ) , πr2 ∗ Lr2 τ (ϕ2 ) ⊗ F (N2 ) ² ³ ¡ ¡ ¢ ¢´ Hom πr1 ∗ Lr1 (τ +1) (ζ1 ) ⊗ F (N1 ) , πr2 ∗ Lr2 (τ +1) (ζ2 ) ⊗ F (N2 ) ,
with ϕi (z) = (t∗xi ϕ0 (ri τ ))(z)ϕ0 (ri τ, z)ni −1 and ζi (z) = t∗x + ri ni ϕ0 (ri (τ + 1), z)ϕ0 (ri (τ + 1), z)ni −1 i
2
and F (Ni ) is short for F (Vi , exp Ni ). Write, as always, r = lcm(r1 , r2 ), ri0 = ri gcd(r1 ,r2 ) . Now on the left hand side we use the following identifications: ³ ¡ ¢´ ¡ ¢ Hom πr1 ∗ Lr1 τ (ϕ1 ) ⊗ F (N1 ) , πr2 ∗ Lr2 τ (ϕ2 ) ⊗ F (N2 ) see 2.14 ∼flat base change
² ³ ¡ ¢ ∗ ¡ ¢´ Ld−1 ∗ Hom π ˜ L (ϕ ) ⊗ F (N ) , π ˜ L (ϕ ) ⊗ F (N ) r τ 1 1 r τ 2 2 1 2 1,ν 2,ν ν=0
³ ´ 0 N ), L (ψ ) ⊗ F (r 0 N ) Hom L (ψ ) ⊗ F (r rτ 1 rτ 2,ν 2 1 1 2 ν=0
Ld−1
³ ´ 0 L (ψ ψ −1 ) ⊗ F (V ⊗ V ∨ , r 0 N − r 0 N ) H rτ 2 2 1 2,ν 1 1 1 2 ν=0
Ld−1
O
νψ,N
³ ´ 0 L (ψ) ⊗ (V ⊗ V ∨ ), H rτ 2 1 ν=0
Ld−1
with the abbreviations 0
ψ1 (z) = t∗r0 x1 ϕ0 (rτ, z)ϕ0 (rτ, z)r2 n1 −1 , 2
0
ψ2,ν (z) = t∗r0 x2 +n2 r0 ντ ϕ0 (rτ, z)ϕ0 (rτ, z)r1 n2 −1 , 1
1
ψ = ψ2,ν ψ1−1 , and N = r10 N2 − r20 N1 .
6.1 The functors corresponding to τ 7→ τ + 1
61
Analogously on the right hand side we have ³ ¡ ¢ ¡ ¢´ Homτ +1 πr1 ∗ L(ζ1 ) ⊗ F (N1 ) , πr2 ∗ L(ζ2 ) ⊗ F (N2 ) see 2.14 ∼flat base change
² ³ ¡ ¢ ∗ ¡ ¢´ ∗ ˜1,ν L(ζ1 ) ⊗ F (N1 ) , π ˜2,ν L(ζ2 ) ⊗ F (N2 ) ν=0 Homr(τ +1) π
Ld−1
³ ´ 0 N ), L(ξ ) ⊗ F (r 0 N ) Hom L(ξ ) ⊗ F (r 1 1 2,ν 2 r(τ +1) 2 1 ν=0
Ld−1
³ ´ −1 0 L ∨ , r0 N − r0 N ) H (ξ ξ ) ⊗ F (V ⊗ V 2 r(τ +1) 2,ν 1 1 1 2 2 1 ν=0
Ld−1
O
νξ,N
³ ´ 0 L H (ξ) ⊗ (V2 ⊗ V1∨ ), r(τ +1) ν=0
Ld−1 with
0
ξ1 = t∗r0 x1 + rn1 ϕ0 (r(τ + 1), z)ϕ0 (r(τ + 1), z)r2 n1 −1 , 2
2
0
ξ2,ν = t∗r0 x2 + rn2 +n2 r0 ν(τ +1) ϕ0 (r(τ + 1), z)ϕ0 (r(τ + 1), z)r1 n2 −1 , 1
ξ=
1
2
ξ2,ν ξ1−1 ,
and N like before. We can calculate b∗ by giving for each component ν the map ¡ ¢ ¡ ¢ H 0 Lrτ (ψ) ⊗ (V2 ⊗ V1∨ ) − → H 0 Lr(τ +1) (ξ) ⊗ (V2 ⊗ V1∨ ). We can pass from Lrτ (ψ) to Lr(τ +1) (ξ) by using item 1 r times. On the V -component b∗ acts as the identity and since ψ(z) = ξ(z), it follows that νψ,N = νξ,N . This gives the desired expression. g. For torsion sheaves we can simply map the support and get the assertion. h. The assertion for morphisms between to and from torsion sheaves follows by definition.
Together with the mirror symmetry functors, b∗ gives us an equivalence of the corresponding Fukaya-categories, Db (Eτ ) b∗
²
Db (Eτ +1 )
Φτ
Φτ +1
/ FK(E τ ) ²
Φb∗
/ FK(E τ +1 ).
62
Automorphisms of the Elliptic Curve
Using the definition of the mirror symmetry-functor, we can simply calculate, what it does to objects and morphisms: Let L = (Λ, α, M ) be an indecomposable object of FK(E τ ). 1 1 α ∈ (2k − , k + ), 2 2 1 or α = 2k + . 2 ¡ ¢ In the second case, L is isomorphic to Φτ S(x, V, M ) , the image of a torsion sheaf; in the¡first case there exist¢ V, N, x = aτ + b, r, n, s.t. L is isomorphic to the image of πr∗ L(ϕ) ⊗ F (V, exp N ) , in the sense we used all these symbols before. (Because Φτ is an equivalence, for each object in FK(E τ ) exists an object in Db (Eτ ) s.t. its image under the equivalence is isomorphic to our original object). For simplicity we first assume that we are in the first case with r = 0. Then we have, with all the conventions as above: L(t∗x ϕ0 ϕn−1 ) ⊗ F (V, exp N ) b∗
L(t∗
nζ ζ x+ 2 0
²
n−1 )
⊗ F (V, exp N )
Φτ
Φτ +1
/ L = (Λ, α, M ) ,
/ L0 = (Λ, α, M 0 )
where Λ = (a + t, a(n − 1) + nt) ¢ ¡ α ∈ − 21 , 21 M = (V, exp(−2πib + N ) ¡ ¢ n M 0 = V, exp(−2πi(b − a + ) + N ) . 2 So what Φb∗ does in this case is keeping the underlying Lagrangian fixed, ¡ ¢ while n twisting the monodromy of the local system by the factor exp 2πi(a − 2 ) . Proposition 6.2. The base change τ 7→ τ + 1 of the lattice, sending thus Eτ to Eτ +1 , induces a nontrivial equivalence Φb∗ of the corresponding Fukaya categories. After an appropriate shift each indecomposable object L of the Fukaya category either has α = 21 or is of the form pr∗ (Λ, α, M) = L, ¡ ¢ where (Λ, α, M) = Φrτ Lrτ (ϕ) ⊗ Frτ (V, exp N ) . Now Φb∗ is the functor ¡ ¡ ¢ ¢ pr∗ (Λ, α, M) 7→ pr∗ Λ, α, exp 2πir(a − n2 ) M , , x = arτ + b. And if α is a multiple of 12 , then with ϕ = t∗x ϕ0 ϕn−1 0 ¡ ¢ L = Φτ S(x, V, N ) , and then
¢ ¡ ¢ ¡ 1 Λ, 2 , M 7→ Λ, 12 , exp(−2πia)M .
6.1 The functors corresponding to τ 7→ τ + 1
63
On morphisms, Φb∗ operates as follows: ³ ¡ ¢´ If Li = (Λ, α, M ) = Φτ πri ∗ L(ϕi ) ⊗ F (Vi , exp Ni ) then gcd(r1 ,r2 )−1
Hom(L1 , L2 ) =
M
Hom(˜ p∗1,ν L1 , p˜∗2,ν L2 ),
(6.1.1)
ν=0
and for T · ek,ν ∈ Hom(˜ p∗1,ν L1 , p˜∗2,ν L2 ) we get with r = ¡ T · ek,ν 7→ exp πirk(1 −
k ) r10 n2 −r20 n1
r1 r2 gcd(r1 ,r2 )
−a1 − 2πi r0 na22 −r 0 n + πir 1 1
2
and ri0 =
ri gcd(r1 ,r2 ) :
− n1 ¢ T · ek,ν . − r20 n1
n2 r10 n2
The only remaining case, in which there are more morphisms than just the zero morphism is the case, where L1 is the symplectic counterpart of a vectorbundle and L2 corresponds to a torsion sheaf. Let ³ ¡ ¢´ L1 = Φτ πr∗ L(t∗x1 ϕ0 ϕ0n−1 ) ⊗ F (V1 , exp N1 ) ¡ ¢ L2 = Φτ S(x2 , V2 , N2 ) Then Hom(L1 , L2 ) =
r M
V2 ⊗ V1∨ = Hom(Φb∗ L1 , Φb∗ L2 ).
k=1
Define a1 , b1 , a2 , b2 by the equations x1 = a1 rτ + b1 , x2 = a2 τ + b2 . Then we have as operation of Φb∗ on V2 ⊗ V1 for the component k: ¡ ¢ 2 n Φb∗ : T 7→ exp πin (a2 +k) + 2πi(a1 − 2r )(a2 + k) T. r Proof. The algebraic counterpart for an indecomposable object L of the Fukaya category is either a torsion sheaf or a vector bundle. Suppose it is the torsion sheaf S(aτ + b, V, N ). Then we have the following diagram: ¡ ¢ S aτ + b, V, N
Φτ
/ (Λ, α, M)   ²
b∗
² ¡ ¢ Φτ +1 / (Λ, α, exp(−2πia)M). S a(τ + 1) + (b − a), V, N ¡ ¢ ˜ If the algebraic counterpart of L ¡ ¢ is πr∗ L(ϕ) ⊗ F (V, exp N ) , then denote by L the object Φrτ L(ϕ) ⊗ F (V, exp N ) . With the usual notation L = (Λ, α, M), and ¡ ¢ Λ = (a + t, a(n − 1) + nt) and M = V, exp(2πib + N ) .
With the base change functor we have the diagram: ¢ Φτ ¡ πr∗ L(t∗x ϕ0 ϕ0n−1 ) ⊗ F (V, exp N ) b∗
² ¡ ¢ πr∗ L(t∗x+ rn ζ0 · ζ0n−1 ) ⊗ F (V, exp N ) 2
³
/ pr∗ (Λ, α, M)   ²
´
¡ ¢ / pr∗ Λ, α, exp 2πi(ar − rn ) M 2
Φr(τ +1)
¡ exp 2πi(ar −
¢
rn 2 )
¡ ¢ · pr∗ Λ, α, M
64
Automorphisms of the Elliptic Curve
¡ ¢ For the morphisms note that for Li = Φτ (πri ∗ L(ϕi ⊗ F (Vi , exp Ni )) ), gcd(r1 ,r2 )
Hom(L1 , L2 ) ∼ =
M
Hom(˜ p∗1,ν L1 , p˜∗2,ν L2 )
ν=0
like in proposition 2.22 diagram 2.4.2. Then we have for a morphism T˜ · ek,ν the algebraic counterpart fk,ν · T with fk,ν like in proposition 6.1 and n ¡ (a2 −a1 + 2 ν)2 ¢ T˜ = exp −πi r0 n2 −rr0 2n1 1 2 ¡ a2 −a1 + nr22 ν 0 b r 0 −b r 0 ¢ exp r0 n2 −r0 n1 (r1 N2 ⊗ 1V1∨ − 1V2 ⊗ r20 N1∨ − 2πi1V2 ⊗V1∨ r02n12 −r10 n21 ) T. 1
2
1
2
Then we have the diagram: Φrτ
fk,ν ⊗ T b∗
¡ exp πirk(1 −
²
k
r10 n2 −r20 n1
¢ ) gk,ν ⊗ T
/ T˜ · e  k,ν  ²
Φr(τ +1)
/ T˜ 0 · e k,ν
with T˜0 = exp
³a
2 2 −a1 + r ¡ 0 2 r10 n2 −r20 n1 r1 N2 n
⊗ 1V1∨ − 1V2 ⊗ r20 N1∨ − 2πi1V2 ⊗V1∨
r10 b2 −r20 b1 −(a2 −a1 )+r r10 n2 −r20 n1
n2 −n1 2
¢´
¡ ¢2 ´ ³ n ¢ ¡ a2 −a1 + r 2 ν k 2 T exp πirk(1 − r0 n2 −r0 n1 ) exp −πi r0 n2 −r0 n1 2 2 1 1 ¡ ¢ ¡ ¢ ¡ ¢ a2 −a1 k 1 ˜ = exp πirk(1 − r0 n2 −r exp πir r0 nn22 −n 0 n ) exp −2πi r 0 n −r 0 n −r 0 n1 T , 1 2 1 1
2
1
2
1
2
as claimed. It remains to proof the assertion for a symplectic counterpart of a torsion sheaf. First not that on the algebraic side we use the following identifications: ³ ¡ ´ ¢ Hom πr∗ L(t∗x1 ϕ0 ϕ0n−1 ) ⊗ F (V1 , exp N1 ) , S(x2 , V2 , N2 ) ∼ =
²
¡ ¢ Hom L(t∗x1 ϕ0 ϕn−1 ) ⊗ F (V1 , exp N1 ), πr∗ S(x2 , V2 , N2 ) 0 ∼ =
² ¡ ∗ ¢ Lr n−1 ) ⊗ F (V1 , exp N1 ), S(x2 + kτ, V2 , N2 ) k=1 Hom L(tx1 ϕ0 ϕ0
∼ =
/ Lr
k=1
b∗
² ¡ ∗ ¢ n−1 Hom L(t ) ⊗ F (V1 , exp N1 ), S(x2 + kτ, V2 , N2 ) n ψ0 ψ 0 x1 + k=1
Lr
2
³
²
V2 ⊗ V1∨ id
∼ =
² ∨ k=1 V2 ⊗ V1
/ Lr
∼ =
´ ¡ ¢ Hom b∗ πr∗ L(t∗x1 ϕ0 ϕ0n−1 ) ⊗ F (V1 , exp N1 ) , b∗ S(x2 , V2 , N2 )
On the symplectic side we have the analogous identifications and we use only the middle square (with the strong arrows) to define Φb∗ . We may do this according
6.1 The functors corresponding to τ 7→ τ + 1
65
k the factor to the definition of Φτ and Φτ +1 (see chapter 2). Now denote with Crτ k from the operation of Φrτ on the k−th component of this sum and by Cr(τ +1) the corresponding factor of the operation of Φr(τ +1) . Then we calculate from the definition with x1 = a1 rτ + b1 , x2 = a2 τ + b2 :
¡ ¢ 2 k + 2a1 (a2r+k) ) − 2πi( (a2r+k) b1 + a1 b2 + n (a2r+k) b2 ) Crτ = exp −πirτ (n (a2r+k) 2 ¢ ¡ exp −N2 ⊗ (a1 + n (a2r+k) )1V1∨ + 1V2 ⊗ (a2r+k) N1∨ ¡ ¡ a2 +k ¢¢ (a2 +k)2 k a2 +k n Cr(τ + 2a ) − 2πi (b − a r + ) 1 1 1 2 +1) = exp −πir(τ + 1)(n r r 2 r ¡ ¡ ¢¢ exp −2πi +a1 (b2 − a2 − k) + n a2r+k (b2 − a2 − k) ¡ ¢ exp −N2 ⊗ (a1 + n a2r+k )1V1∨ + 1V2 ⊗ a2r+k N1∨ . The morphism Φb∗ is given by ¢ ¡ 2 k k −1 + 2a1 a2r+k ) Cr(τ = exp −πir(n (a2r+k) 2 +1) · (Crτ ) ³ ¡ ¢´ · exp −2πi a2r+k (−a1 r + n2 ) + a1 (−a2 − k) + n a2r+k (−a2 − k) ¡ ¢ 2 n = exp πin (a2 +k) + 2πi(a1 − 2r )(a2 + k) . r
Remark 6.3. We see that b∗ has the symplectic counterpart G, which leaves the underlying Lagrangians fixed and twists the local systems. In chapter 9, we will generalize our morphisms, and equip them with a line bundle, such that twisting the local systems will become an automorphism of FKE τ . But we will not get a functor FKE τ − → FKE τ +1 , because if we choose to define as morphisms between the underlying manifolds complex symplectic maps, then there are no morphisms between these manifolds. Neither are there complexified Lagrangian correspondences that we could equip with such a local system. If we define as morphisms of the underlying manifolds symplectic maps that only respect the imaginary i.e. symplectic part of the complexified symplectic form, then the manifolds are equal (isomorphic by means of the identity). However, the identity also induced the identity on the morphisms spaces, such that this cannot yield G. Also an modification of the functor induced by the identity by composing with the functor that tensors with restrictions of a local system given on Eτ cannot yield G, because the factor exp(2πir(a − n2 )) is not of the form mnx · mry with mx and my holomorphic maps, independent of a specific object. (Compare chapter 9 about how functors obtained by such a construction look like.) Therefore also in this case we find no symplectic counterpart which we understand intrinsically.
6.1.2
The pullback and its symplectic counterpart
Since b is an isomorphism b∗ and b∗ are not only adjoint, but inverse to each other. And further since b∗ = (b∗ )−1 = (b−1 )∗ we have intrinsically also described the pullback along the base change τ 7→ τ − 1. Proposition 6.4. The base change τ 7→ τ + 1 induces the functor b∗ : Db (Eτ ) −→ Db (Eτ +1 ). It is given by:
66
Automorphisms of the Elliptic Curve 1. An indecomposable vector bundle is given by πr∗ (L(ζ) ⊗ F ), where now L(ζ) ⊗ F is a special vector bundle over Er(τ +1) with ζ0 (z) = exp(−πir(τ + 1) − 2πiz) = ϕ0 (r(τ + 1), z) and ϕ0 = exp(−πirτ + 2πiz) = ϕ(rτ, z). The base change is given by: πr∗
¡
Ob(Db Eτ ) −→ Ob(Db Eτ +1 ) ¢ ¡ ¢ L(t∗x ζ0 · ζ0n−1 ) ⊗ F (V, exp N ) 7→ πr∗ L(t∗x− rn ϕ0 · ϕn−1 ) ⊗ F (V, exp N ) , 0 2
2. Recall that morphisms in the case ¡ ¢ Ai = πri L(ϕi ) ⊗ F (V, exp Ni ) can be viewed (see 2.14) as a direct sum of morphisms between objects on Erτ , r1 r2 : where r = gcd(r 1 ,r2 ) ³ ¡ ¢ ¡ ¢´ Hom πr1 ∗ Lr1 (τ +1) (ϕ1 ) ⊗ Fr1 (τ +1) (V1 , exp N1 ) , πr2 ∗ Lr2 (τ +1) (ϕ2 ) ⊗ Fr2 (τ +2) (V2 , exp N2 ) ∼ =
gcd(r1 ,r2 )
M
0 r 0 n1 −r1 n2 −1
H 0 (L(t∗w(ν) )ζ0 ζ0 2
) ⊗ V2 ⊗ V1∨ ,
ν=1
and we get ¡ gk,ν ⊗ T 7→ exp − πirk(1 −
¢ fk,ν ⊗ T,
k n2 −n1 )
for each of the components of the direct sum.
fk,ν = t∗
ζ0 (z) = ϕ0 (rτ, z), 0 y(ν) = r1 x2 − r20 x1 + n2 r10 ντ, w(ν) = r10 (x2 + r22n2 ) − r20 (x1 + r12n1 ) + n2 r10 ν(τ + 1) ¡ 0 ¢ 0 0 0 k θ[ r0 n2 −r 0 n , 0] (r1 n2 − r2 n1 )rτ, (r1 n2 − r2 n1 )z y(ν) 1 1
0 n −r 0 n r1 2 2 1
gk,ν = t∗
w(ν) 0 n −r 0 n r1 2 2 1
2
¡ 0 ¢ 0 k θ[ r0 n2 −r 0 n , 0] (r1 n2 − r2 n1 )r(τ + 1), (n2 − n1 )z 1 1
2
3. for Torsion sheaves: Ob(Db Eτ +1 ) −→ Ob(Db Eτ ) S(x, V, N )
7→
S(x, V, N )
4. For two torsion sheaves with different support, the spaces of morphisms are zero anyway, 5. We still have to view the cases of Hom(A1 , A2 ), where either A1 or A2 is a torsion sheaf. If A1 is a torsion sheaf, this is the trivial group. If A1 is not a torsion sheaf, but A2 is, then Hom(A1 , A2 ) = V2 ⊗ V1∨ does not depend on τ and b∗ is the identity morphism.
6.1 The functors corresponding to τ 7→ τ + 1
67
Proof. For a line bundle, we have to show: Ob(Db Eτ +1 ) − → Ob(Db Eτ ) Lτ +1 (t∗x ζ0 · ζ0n−1 )
Lτ (t∗x− n ϕ0 · ϕ0n−1 )
7→
2
We have calculated in the proof of proposition 6.1: ϕ = t∗n ζ. 2
Now we use t−1 x = t−x and see:
ζ = t∗− n ϕ. 2
Again we can proceed step by step and all other claims are in the same way analogous to the case of pullback. Proposition 6.5. The base change τ 7→ τ + 1 of the lattice, sending thus Eτ to Eτ +1 , induces a nontrivial equivalence Φb∗ of the corresponding Fukaya categories, which is given by the following maps. After a shift, each indecomposable object L of the Fukaya category either has α = 21 or is of the form pr∗ (Λ, α, M ) = L, ¡ ¢ where (Λ, α, M ) = Φr(τ +1) Lrτ (ϕ) ⊗ Frτ (V, exp N ) . Now Φb∗ is the functor ¡ ¢ pr∗ (Λ, α, M) 7→ pr∗ (Λ, α, exp −2πir(a − n2 ) M), with ϕ = t∗x ϕ0 ϕn−1 , x = ar(τ + 1) + b. And if α is a multiple of 12 , then 0 ¡ ¢ L = Φτ +1 S(x, V, N ) , and then (Λ, 12 , M) 7→ (Λ, 12 , exp(2πia)M). On morphisms Φb∗ operates as¡ follows: ¢ If Li = (Λ, α, M ) = Φτ +1 (πri ∗ L(ϕi ) ⊗ Fi then gcd(r1 ,r2 )−1
Hom(L1 , L2 ) =
M
Hom(˜ p∗1,ν L1 , p˜∗2,ν L2 ),
(6.1.2)
ν=0
and for T · ek,ν ∈ Hom(˜ p∗1,ν L1 , p˜∗2,ν L2 ) we get with r = ¡ T · ek,ν 7→ exp −πirk(1 −
k ) r10 n2 −r20 n1
r1 r2 gcd(r1 ,r2 )
−a1 + 2πi r0 na22 −r 0 n − πir 1 1
2
and ri0 =
n2 0 r1 n2
ri gcd(r1 ,r2 ) :
− n1 ¢ T · ek,ν . − r20 n1
The only remaining case, in which there are more morphisms than just the zero morphism is the case, where L1 is the symplectic counterpart of a vectorbundle and L2 corresponds to a torsion sheaf. Let ζ0 = ϕ0 (r(τ + 1), z) and ³ ¡ ¢´ L1 = Φτ πr∗ L(t∗x1 ζ0 ζ0n−1 ) ⊗ F (V1 , exp N1 ) ¡ ¢ L2 = Φτ S(x2 , V2 , N2 )
68
Automorphisms of the Elliptic Curve
Then Hom(L1 , L2 ) =
r M
V2 ⊗ V1∨ = Hom(Φb∗ L1 , Φb∗ L2 ).
k=1
Define a1 , b1 , a2 , b2 by the equations x1 = a1 r(τ + 1) + b1 , x2 = a2 (τ + 1) + b2 . Then we have as operation of Φb∗ on V2 ⊗ V1 for the component k: ¡ ¢ 2 n Φb∗ : T 7→ exp πin (a2 +k) − 2πi(a1 + 2r )(a2 + k) T. r Remark 6.6. The question, whether we have a geometric interpretation for Φb∗ naturally has the same answer like for the pushforward (see 6.3): We definitely have no symplectic counterpart in FKell, since the functors does not come from a morphism of the underlying manifolds. And we will also not find one by extending our notion of morphism in the category of Fukaya categories in chapter 9.
6.2
The functors corresponding to τ 7→ − τ1
Now we investigate the existence of a symplectic counterpart of the morphism, τ 7→ − τ1 . We proceed as follows: First, we calculate the pullback along this morphism and its symplectic counterpart. Then we calculate the pushforward and the corresponding symplectic functor. We cannot expect the counterparts to lie in pFKell, since there is no morphism of the underlying manifolds. To τ 7→ − τ1 there correspond two elements of SL2 (Z): J and −J = J 3 , where µ ¶ 0 1 J= . −1 0 1. to J corresponds
C Zτ +Z
/
C Z+Z(−τ )
− τ1
/
/ −z τ
z 2. to −J corresponds
C Z(− τ1 )+Z
C Zτ +Z
/
1 τ
C Z(−1)+Zτ
/
C Z(− τ1 )+Z
/ z
z
τ
The two possibilities differ by a multiplication by (−1), which is an automorphism of the Riemann surface C/(Z(− τ1 ) + Z). It induces a natural autoequivalence of Db (E− 1 ), that was already described in chapter 4 and that has a symplectic τ counterpart. For the remainder of this chapter we stick to J and denote the map Eτ − → E− 1 τ
by m. Since m is an isomorphism m∗ = (m∗ )−1 = (m−1 )∗ and m−1 corresponds to J −1 = J 3 = −J. First we investigate, how m∗ and m∗ operate on line bundles.
6.2 The functors corresponding to τ 7→ − τ1
69
Lemma 6.7. Let m denote the map Eτ −→ E− 1 , τ
given by z z 7→ − . τ Let further ζ0 denote the function ζ0 (z) = exp(−πi(− τ1 ) + 2πiz) = ϕ0 (− τ1 , z) and as usual, ϕ0 (z) = ϕ0 (τ, z). Then ¡ ¢ m∗ L(t∗y ζ0 ζ0n−1 ) ∼ ) = L(t∗−yτ ϕ0 ϕn−1 0 ¡ ∗ ¢ n−1 n−1 ∗ m∗ L(tx ϕ0 ϕ0 ) ∼ = L(t− x ζ0 ζ0 ).
(6.2.1) (6.2.2)
τ
Proof. The inverse and direct image of the line bundles L(ϕ0 ) and L(ζ0 ) are easily determined by using the fact that sending a line bundle to its corresponding divisor 1+τ 2
is functorial. Just notice that the divisor corresponding to L(ϕ0 ) is for L(ζ0 ). Since ¢ ¡ 1 m 1+τ 2 − x = − 2τ −
1 2
+
x τ
≡
1+(− τ1 ) 2
and
1− τ1 2
− (− xτ ),
we have m∗ L(t∗x ϕ0 ) ∼ = L(t∗− x ζ0 ) and m∗ L(t∗y ζ0 ) ∼ = L(t∗−yτ ϕ0 ). τ
A different way to calculate inverse and direct image is by working with bundle maps. This has the advantage that we know how to map sections of bundles, in our case this means we know immediately how the functors operate on the morphism spaces. m∗ L(ζ) is the line bundle F (C, 1, ζ ◦ m), defined by the multipliers z e1 = ζ(− ), τ
and eτ = 1
and m∗ L(ϕ) is defined by multipliers e− 1 = 1 τ
and e1 (z) = ϕ(−zτ − z)−1 .
Note that ( )−1 denotes the multiplicative inverse, not the inverse function. Of course these are isomorphic to the bundles of lemma 6.7 just by construction. Lemma 6.8. The following bundles are isomorphic: ¢ ¡ ¡ ¢ ) 1. m∗ L(t∗y ζ0 ζ0n−1 ) = F C, 1, (t∗y ζ0 ζ0n−1 ) ◦ (− τz ) ∼ = L(t∗−yτ ϕ0 ϕn−1 0 ¢ ¢ ¡ ¡ ) ◦ (−zτ − τ )−1 ∼ 2. m∗ L(t∗x ϕ0 ϕn−1 ) = F C, 1, (t∗x ϕ0 ϕn−1 = L(t∗− x ζ0 ζ0n−1 ) 0 0 τ
In the following we make these isomorphisms explicit.
70
6.2.1
Automorphisms of the Elliptic Curve
The pullback and its symplectic counterpart
To see how morphisms of line bundles are mapped under m∗ we need to know the isomorphisms of lemma 6.8. ¡ ¢ ∼ F C, 1, (t∗y ζ0 ζ0n−1 )(− τz ) − → L(t∗−yτ ϕ0 ϕn−1 ). 0 Consider g(z) = exp(2πiyz − πi nτ z 2 ). It is immediate that g is holomorphic in z. For g to be an isomorphism between these bundles it has to fulfill the following relations: ¡ ¢−1 g(z + 1) = t∗y ζ0 ζ0n−1 (− τz ) g(z) g(z + τ ) = (t∗−yτ ϕ0 ϕn−1 )(z)g(z). 0 And the following calculations show that these are really satisfied. g(z + 1) = exp(2πiy(z + 1) − πi nτ (z + 1)2 ) ¡ ¢ = exp 2πiy(z + 1) − πi nτ (z 2 + 2z + 1) ³ ¡ ¢´−1 = exp(2πiyz − πi nτ z 2 ) exp −2πiy − πi(− nτ ) − 2πi(− nz ) τ ¡ ∗ n−1 z ¢−1 = g(z) ty ζ0 ζ0 (− n ) g(z + τ ) = exp(2πiy(z + τ ) − πi nτ (z + τ )2 ) ¡ ¢ = exp 2πiy(z + τ ) − πi nτ (z 2 + 2zτ + τ 2 ) = exp(2πiyz − πi nτ z 2 ) exp(2πiyτ − πinτ − 2πinz) = g(z)t∗−yτ ϕ0 ϕn−1 (z). 0 Lemma 6.9. Global sections of L(t∗y ζ0 ζ0n−1 ) are transformed under m∗ into global sections of L(t∗−yτ ϕ0 ϕn−1 ) according to the formula 0 ∗
1 t y θ[ m n , 0](n(− τ ), nz) n
7→
τ 12 exp(− πi 4 )( n )
2 exp(πi ynτ )
n−1 X
∗ exp(2πi m n k)t
k=0
1
yτ − n
θ[ nk , 0](nτ, nz).
For the root ( nτ ) 2 choose the solution that lies in the first quadrant of the complex plane. ¡ ¢ 1 Proof. We start with t∗y θ[ m n , 0] n(− τ ), nz =: hm , an element of our standard basis n
of the space of global sections of L(t∗y ζ0 ζ0n−1 ). Then 1 nz m∗ hm = t∗− yτ θ[ m n , 0](n(− τ ), − τ ), n
is a global section of ¢ ¡ m∗ L(t∗y ζ0 ζ0n−1 ) = F C, 1, (t∗y ζ0 ζ0n−1 )(− τz )
6.2 The functors corresponding to τ 7→ − τ1
71
Notice that by definition 2
1 nz m mz m n nz t∗− yτ θ[ m n , 0](n(− τ ), − τ ) = exp(−πi nτ − 2πi τ + 2πi n y)θ[0, 0](− τ , − τ + y − n
m τ )
(in the following we simply write θ(τ, z) for θ[0, 0](τ, z)) and according to the transformation formula A.8 this equals 1
2
2
y τ 2 nz τ = exp(− πi 4 )( n ) exp(πi τ + πi n τ − 2πiyz)θ( n , z +
m n
−
yτ n )
And it follows that 1 nz t∗− yτ θ[ m n , 0](n(− τ ), − τ )g(z) = n
1
2
2
y τ nz n 2 τ 2 = exp(− πi 4 ) exp(πi τ + πi n τ − 2πiyz + 2πiyz − πi τ z )( n ) θ( n , z + 2
1
y τ τ 2 τ = exp(− πi 4 ) exp(πi n )( n ) θ( n , z +
m n
−
m n
−
yτ n )
yτ n )
Now we apply the formula from lemma A.6 and get 1 nz t∗− yτ θ[ m n , 0](n(− τ ), − τ )g(z) = n
2
1
2
1
n−1 X
1
k=0 n−1 X
y τ τ 2 τ = exp(− πi 4 ) exp(πi n )( n ) θ( n , z + y τ τ 2 = exp(− πi 4 ) exp(πi n )( n )
2
y τ τ 2 = exp(− πi 4 ) exp(πi n )( n )
m n
−
yτ n )
θ[ nk , 0](nτ, nz − yτ + m) exp(2πi nk m)t∗
k=0
yτ − n
θ[ nk , 0](nτ, nz)
Lemma 6.10. Let F1 , F2 be two special vector bundles, i.e. Fi = L(t∗yi ζ0 ζ0ni −1 ) ⊗ F (Vi , exp Ni ). ¡ ¢ Then Hom(F1 , F2 ) ∼ = H 0 L(ζ) ⊗ V with ζ = t∗y2 −y1 ζ0 ζ0n2 −n1 −1 and V = V2 ⊗ V1∨ . With this notation ¡ ¢ n −1 1. m∗ Fi ∼ = L(t∗−yi τ ϕ0 ϕ0 i ) ⊗ F Vi , exp(−τ N ) 2. For n2 > n1 , morphisms are mapped as ¡ ¢ ¡ ¢ H 0 L(ζ) ⊗ V −→ H 0 L(ϕ) ⊗ V given by n t∗y θ[ m n , 0](− τ , nz) ⊗ T n _
² k τN2 πi τ 12 τ 2 k ∗ )( ) y + 2πi m)t exp(− exp(πi k=0 − yτ θ[ n , 0](nτ, nz) ⊗ exp( 4πin − 4 n n n
Pn−1
n
τN n y)T.
72
Automorphisms of the Elliptic Curve
Proof.
1. First of all we can write the bundles Fi as L(t∗yi ζ0 · ζ0ni −1 ) ⊗ F (Vi , exp Ni ) = F− 1 (Vi , t∗yi ζ0 · ζ0ni −1 exp Ni , 1), τ
and then we calculate m∗ F− τ1 (Vi , t∗yi ζ0 · ζ0ni −1 exp Ni , 1) = Fτ (Vi , 1, (t∗yi ζ0 · ζ0ni −1 )(− τz ) exp Ni ) ¡ ¢ = Fτ C, 1, (t∗yi ζ0 · ζ0ni −1 )(− τz ) ⊗ Fτ (Vi , 1, exp Ni ) ∼ Lτ (t∗ ϕ · ϕni −1 ) ⊗ Fτ (Vi , exp(−τ Ni )) = −yi τ
0
0
The isomorphism ¡ ¢ g : Fτ C, 1, (t∗yi ζ0 · ζ0ni −1 )(− τz ) − → Lτ (t∗−yi τ ϕ0 · ϕn0 i −1 ) was explicitly given above and the isomorphism ¡ ¢ Fτ (Vi , 1, exp Ni ) − → Fτ Vi , exp(−τ Ni ) is given by (z, v) 7→ (z, exp(−zNi )v). d 2. First note that exp(N dz )f (z) = f (z + N ), for an analytic function f and a matrix N . The notation f (z + N ) is defined by inserting the matrix z 1 + N into the powerseries of f . Then we can make calculations similar to those in the previous lemma. We use the notation N = N2 ⊗ 1V1 − 1V2 ⊗ N1∨ , y = y2 − y1 , ζ = ζ2 ζ1−1 , ϕ = ϕ2 ϕ−1 1 , n = n2 − n1 . Consider the following diagram. We are interested in the morphism denoted by the broken line:
¡ ¢ H 0 F (V, ζ exp N ) o
νζ,N
¡ ¢ H 0 L(ζ) ⊗ V Â Â
m∗
Â
² ³ ¡ ¢´ 0 H F V, 1, (ζ exp N )(− τz ) ²
H0
   Â
G
³ ¡ ¢´ F V, ϕ exp(−τ N ) o
Â
νϕ,−τ N
H0
¡ ² ¢ L(ϕ) ⊗ V
The arrow G is given by the function G(z) = exp(2πiyz − πi nτ z 2 − N z), which is obviously a holomorphic isomorphism between the two bundles. Then we use this diagram to calculate just like before: n t∗y θ[ m n , 0](− τ , nz) ⊗ T n
is mapped by νζ,N to ¡ ¢∗ m N d exp (− 2πin ) dz t y θ[ n , 0](− nτ , nz) · T n
n = θ[ m n , 0](− τ , nz + y −
N 2πi )
·T
6.2 The functors corresponding to τ 7→ − τ1
73
by m∗ this maps to n nz θ[ m n , 0](− τ , − τ + y −
N 2πi )
·T
we apply G and get n nz exp(2πiyz − πi nτ z 2 − N z)θ[ m n , 0](− τ , − τ + y −
= exp(2πiyz − πi nτ z 2 − N z −
N 2πi ) · T 2 mN πi mn τ − 2πi m ny− n ) N θ(− nτ , − nz τ + y − 2πi
−
m τ )
·T
now we use the formula from A.8 1
τN2 τN τ τ τN m 4πin − n y)θ( n , z − n y + 2πin + n ) · T τN τN2 τN exp(−πi nτ y 2 )θ( nτ , z − nτ y + 2πin +m n ) · exp( 4πin − n y)T
τ 2 τ 2 = exp(− πi 4 )( n ) exp(−πi n y + 1
τ 2 = exp(− πi 4 )( n ) −1 applying νϕ,−τ N yields 1
τ 2 τ τ τ 2 exp(− πi 4 )( n ) exp(−πi n y )θ( n , z − n y +
=
n−1 X k=0
2
m n)
τN ⊗ exp( 4πin −
τN n y)T
1
τ 2 τ 2 k exp(− πi 4 )( n ) exp(−πi n y + 2πi n m)
t∗
yτ − n
2
τN θ[ nk , 0](nτ, nz) ⊗ exp( 4πin −
τN n y)T
Before we can pass to considering arbitrary preferred vector bundles, we need another lemma. Lemma 6.11. Let j denote the isomorphism from lemma 2.13, given by j : Es(− 1 ) × Z/dZ −→ Es1 (− 1 ) ×E− 1 Es2 (− 1 ) τ
τ
([z]s(− 1 ) , ν) 7→ ([z]s1 (− 1 ) , [z + τ
τ
τ
τ
ν(− τ1 )]s2 (− 1 ) ), τ
and let i be the isomorphism E τs × Z/dZ −→ E sτ ×Eτ E sτ 1
2
([z] τs , ν) 7→ ([s02 z] sτ , [s01 z − 1
ν τ s2 ] s2 ).
For s1 , s2 ∈ N define: d = gcd(s1 , s2 ) si s0i = d s1 s2 s = lcm(s1 , s2 ) = = s01 s2 = s1 s02 . d
74
Automorphisms of the Elliptic Curve
The following diagram is commutative (including the dotted arrows). a _ ] \ Z X f d b m τ ×id W U g s it
Es(− 1 ) × Z/dZ
E τs × Z/dZ
τ
j
²
m π ˜2
Es1 (− 1 ) ×E− 1 Es2 (− 1 ) τ
τ
τ
²
/ Es2 (− 1 )
τ
q _ X
τ
π ˜1
Es1 (− 1 )
τ s2
²
E sτ o
γ ˜2
2
πs2
²
i
E sτ ×Eτ E sτ 2
1
γs2
²
γ ˜1
² γ1 / E− 1 Xl mτ o Eτ E τ f _ k s1 τ j U πs1 i W X g Z \ ] _ a b d f m
τ s1
Proof. It is easily checked that the diagram commutes. Also Es1 (− 1 ) ×E− 1 Es2 (− 1 ) = m sτ (E sτ ) ×mτ (Eτ ) m sτ (E sτ ), τ
τ
τ
2
2
1
1
which gives us the commutative subdiagram m×id
Es(− 1 ) × Z/dZ o τ
²
E τs × Z/dZ ,
j
i
²
Es1 (− 1 ) ×E− 1 Es2 (− 1 ) o m×m E τ ×Eτ E τ s s τ
τ
τ
2
1
where all the morphisms except i are known to be isomorphisms. It follows that i is an isomorphism. Now we can wrap it all up in the following proposition. Proposition 6.12. The change of lattices τ 7→ − τ1 induces the following functor m∗ : Db (E− 1 ) −→ Db (Eτ ). Consider the vector bundle τ
¡ ¢ πs∗ L(η) ⊗ F (V, exp N ) , where L(η) ⊗ F (V, exp N ) is a special vector bundles over Es(− 1 ) with η0 (z) = τ
exp(−πis(− τ1 ) − 2πiz) = ϕ0 (s(− τ1 ), z). Define further ψ0 = exp(−πi τs + 2πiz) = ϕ0 ( τs , z). Then m∗ acts by Ob(Db E− 1 ) −→ Ob(Db Eτ ) τ ¡ ∗ ¢ ¡ ¢ n−1 πs∗ L(ty η0 · η0 ) ⊗ F (V, exp N ) 7→ γs∗ L(t∗− yτ ψ0 ψ0n−1 ) ⊗ F (V, exp − τs N ) . s
The morphism γs was defined in section 5 to be the map: E τs −→ Eτ , given by z 7→ sz. For torsion sheaves m∗ acts by Ob(Db E− 1 ) −→ Ob(Db Eτ ) τ
S(y, V, N ) 7→ S(−yτ, V, N )
6.2 The functors corresponding to τ 7→ − τ1
75
For morphisms between arbitrary preferred vector bundles we have to find the map ³ ¡ ¢ ¡ ¢´ Hom πs1 ∗ L(η1 ) ⊗ F (N1 ) , πs2 ∗ L(η2 ) ⊗ F (N2 ) ² ³ ¡ ¢ ¡ ¢´ τ Hom γs1 ∗ L(ψ1 ) ⊗ F (− s1 N1 ) , γs2 ∗ L(ψ2 ) ⊗ F (− sτ2 N2 )
The usual identifications according to Proposition 2.14 yield d−1 M
d−1 M ¡ ¢ ¡ ¢ ∗ H 0 L(ξ2,µ ξ1−1 ) ⊗ (V2 ⊗ V1∨ ) −→ H 0 L(χ2,µ χ−1 1 ) ⊗ (V2 ⊗ V1 )
µ=0
µ=0
fk,µ ⊗ T
7→
n−1 X
1
2 τ 2 τ k exp(− πi 4 )( sn ) exp(−πi sn y(µ) + 2πi n m)gk,µ
k=0
⊗ exp
¡
τN2 4πins
−
¢
τN sn y(µ)
T
where we used the following abbreviations: F (Ni ) = F (Vi , exp Ni ) n = s01 n2 − s02 n1 s0i
=
N=
si d r10 N2
d = gcd(s1 , s2 ) s=
⊗ 1V1∨ − 1V2 ⊗ r20 N1∨
y(ν) = s01 y2 − s02 y1 −
n2 s01 ν τ
ni −1 ηi = t∗yi η0,i η0,i , ni −1 y ψ ψ − si 0,i 0,i
ψi = t∗
s0 n −1 t∗s0 y1 ξ0 ξ02 1 2 s01 n2 −1 ξ ξ 1 0 0 s1 y2 −n2 s01 µ(− τ ) 0 s n −1 t∗− y1 τ χ0 χ02 1
ξ2,µ = t∗0 χ1 =
¡ ¢ η0,i = ϕ0 si (− τ1 ) ψ0,i = ϕ0 ( sτi )
i
ξ1 =
s1 s2 d
¡ ¢ ξ0 = ϕ0 s(− τ1 )
s1
s0 n2 −1
χ2,µ = t∗− y2 τ − n2 µ χ0 χ01 s2
s2
χ0 = ϕ0 ( τs )
1 fm,µ = t∗y(µ) θ[ m n , 0](ns(− τ ), nz) n
gk,µ = t∗
−
y(µ)sτ n
θ[ nk , 0](n τs , nz)
For two torsion sheaves with different support, the spaces of morphisms are zero anyway, otherwise we have just the identity. If A1 is a torsion sheaf, Hom(A1 , A2 ) is the trivial group for all A2 6= A1 , as well as the morphism group between the corresponding symplectic counterparts. If A1 is not a torsion sheaf, but A2 is, then Hom(A1 , A2 ) = V2 ⊗ V1∨ does not depend on τ and m∗ is the identity morphism. Proof. The statement about torsion sheaves follows simply from mapping the support.
76
Automorphisms of the Elliptic Curve
Note that the diagram below is commutative. E1τ
m
r
/ Er(− 1 )
²
²
τ
γr
πr m
Eτ
/ E− 1 τ
¡ ¢ ¡ ¢ Therefore m∗ πr∗ (L(ζ) ⊗ F ) ∼ = γr∗ m∗ (L(ζ) ⊗ F ) . So this yields ¡ ¢ ¡ ¢ m∗ πs∗ (L(η) ⊗ F ) ∼ = γs∗ m∗ (L(η) ⊗ F ) ³ ¡ ¢´ ∼ = γs∗ L(t∗− yτ ψ0 ψ0n−1 ) ⊗ F V, exp(− τs N ) . s
For the second part of the proposition we will proceed like this a. the left hand side ³ ¡ ¢ ¡ ¢´ Hom πs1 ∗ L(η1 ) ⊗ F (N1 ) , πs2 ∗ L(η2 ) ⊗ F (N2 ) ∼ =
d−1 M
¡ ¢ H 0 L(ξ2,µ ξ1−1 ) ⊗ V2 ⊗ V1∨ ,
µ=0
b. the right hand side ¡ ¡ ¡ ¢ ¢¢ Hom γs1 ∗ L(ψ1 ) ⊗ F (− sτ1 N1 ) , γs2 ∗ L(ψ2 ) ⊗ F (− sτ2 N2 ) ∼ =
d−1 M
¡ ¢ ∨ H 0 L(χ2,µ χ−1 1 ) ⊗ V2 ⊗ V1
µ=0
c. and then we have to show that m∗ is given by the above formula. We have the usual base change diagram Es(− 1 ) × Z/dZ τ
j
²
Es1 (− 1 ) ×E− 1 Es2 (− 1 ) τ
τ
π ˜1
π ˜2
τ
²
/ Es2 (− 1 ) τ ²
Es1 (− 1 )
πs2
τ
πs2
/ E− 1 τ
π ˜i,µ are defined as restrictions of π ˜i to Es(− 1 ) × {µ}. Using flat base change we get τ
¡ ¡ ¢ ¡ ¢¢ Hom πs1 ∗ L(η1 ) ⊗ F (N1 ) , πs2 ∗ L(η2 ) ⊗ F (N2 ) d−1 ³ M ¡ ¢ ∗ ¡ ¢´ ∗ ∼ Hom π ˜1,µ L(η1 ) ⊗ F (N1 ) , π ˜2,µ L(η2 ) ⊗ F (N2 ) . = µ=0
6.2 The functors corresponding to τ 7→ − τ1
77
The calculation of this pullback can be¡ done directly on ¢ the fibre. As always, ∗ s = lcm(s1 , s2 ), s0i = gcd(ssi2 ,s2 ) . For π ˜1,µ L(η1 ) ⊗ F (N1 ) this yields as multiplier es(− 1 ) = es0 s1 (− 1 ) for the pullback (note: π ˜1,µ = π ˜1,µ0 ∀µ, µ0 ) 2
τ
τ
0
es(− 1 ) τ
2 −1 ³sY ´ ¢ n1 −1 ¡ = (t∗y1 η0,1 η0,1 ) z + ks1 (− τ1 ) · (exp N1 )
k=0 0 ³s2 −1
=
Y
¡ ¢´ 0 exp −2πiy1 + πi sτ1 n1 − 2πin1 (z + s1 (− τ1 )k) exp(N1 )s2
k=0
= exp(−2πis02 y1 + πi τs n1 s02 − 2πin1 s02 z) exp(s02 N1 ) s0 n1
=t∗s0 y1 ξ0 ξ02 2
exp(s02 N1 )
=ξ1 exp(s02 N1 ) ¡ ¢ ∗ The analogous calculation for π ˜2,µ L(η2 ) ⊗ F (N2 ) yields: 0
es(− 1 ) τ
1 −1 ³sY ´ ¢ n2 −1 ¡ (t∗y2 η0,2 η0,2 ) z + ks2 (− τ1 ) + µ(− τ1 ) · (exp N2 ) =
k=0 0
1 −1 ³sY ¡ ¢´ 0 exp −2πiy2 + πi sτ2 n2 − 2πin2 (z + ks2 (− τ1 ) + µ(− τ1 )) · (exp N2 )s1 =
k=0
¢ ¡ = exp(−2πis01 y2 + n2 ν(− τ1 ) + πi τs n2 s01 − 2πis01 n2 z) exp(s01 N2 ) =t∗s0 y
1 0 1 2 +s1 n2 µ(− τ )
n s01 −1
ξ0 ξ0 2
exp(s01 N2 )
=ξs,µ exp(s01 N2 ). With this we have established the isomorphism ¡ ¢¢ ¡ ¢ ¡ Hom πs1 ∗ L(η1 ) ⊗ F (N1 ) , πs2 ∗ L(η2 ) ⊗ F (N2 ) ∼ =
d−1 M
¡ ¢ Hom L(ξ1 ) ⊗ F (s02 N1 ), L(ξ2,µ ) ⊗ F (s01 N2 )
µ=0
∼ =
d−1 M
¡ ¡ ¢´ H 0 L(ξ2,µ ξ1−1 ) ⊗ F V2 ⊗ V1∨ , exp(s01 N1 − s02 N1 )
µ=0
and by applying the isomorphism νξ2,µ ξ−1 ,s0 N1 −s0 N1 that maps 1
¡ ¢ H 0 L(ξ2,µ ξ1−1 ) ⊗ V2 ⊗ V1∨
1
2
³ ¡ ¢´ − → H 0 L(ξ2,µ ξ1−1 ) ⊗ F V2 ⊗ V1∨ , exp(s01 N1 − s02 N1 )
we get the desired expression. The the second step we now want to show ³ ¡ ¢ ¡ ¢´ Hom γs1 ∗ L(ψ1 ) ⊗ F (− sτ1 N1 ) , γs2 ∗ L(ψ2 ) ⊗ F (− sτ2 N2 ) ∼ =
d−1 M µ=0
¡ ¢ ∨ H 0 L(χ2,µ χ−1 1 ) ⊗ V2 ⊗ V1 .
78
Automorphisms of the Elliptic Curve
First note that the bundles L(ψi ) ⊗ F (− sτi Ni ) = m∗ (L(ηi ) ⊗ Ni ) are in fact given by F (C, 1, ηi ◦ m) ⊗ F (Vi , 1, exp Ni ), that is, before we applied the isomorphism to represent them as preferred vector bundles (see the proof of lemma 6.10). We use this representation of the isomorphism class, because we can then calculate the morphism space in analogy to the left hand side. We use a base change diagram like above: E τs × Z/dZ i
²
E sτ ×Eτ E sτ 1
γ ˜2
γ ˜1
/Eτ
s2
2
γs2
²
E sτ
1
γs1
² / Eτ
Define the morphisms γ˜i,µ as the restrictions of γ˜i to E τs × {µ}. Together with flat base change we have achieved so far: ³ ¡ ¡ ¢ ¢´ Hom γs1 ∗ L(ψ1 ) ⊗ F (− sτ1 N1 ) , γs2 ∗ L(ψ2 ) ⊗ F (− sτ2 N2 ) ∼ =
d−1 M
³ ¢ ∗ ¡ ¢´ ¡ ∗ F (V1 , 1, η2 (− zsτ 2 ) exp(N2 )) Hom γ˜1,µ F (V1 , 1, η1 (− zsτ 1 ) exp(N1 )) , γ˜2,µ
µ=0
The next step is to calculate these pullbacks. By definition γ˜1,µ does not depend on µ. We can do the calculation on the fibres and get as multipliers e τs = 1 and the nontrivial multiplier e1 is given by: 0
2 −1 ´ ³sY (s0 z+k)s n1 e1 = (t∗y1 η0,1 η0,1 )(− 2 τ 1 ) exp(N1 )
k=0 0
=
2 −1 ³sY
exp(−2πiy1 + πi sτ1 n1 + 2πin1 s1
´
s02 z+k τ )
0
exp(N1 )s2
k=0
= exp(−2πis02 y1 + πin1 τs s02 + 2πin1 s02 s τz ) exp(s02 N1 ) and for the multipliers of the pullback under γ˜2,µ we get e τs = 1 and for e1 0
1 −1 ³sY ´ (s01 z− sµ +k)s2 n2 2 e1 = (t∗y2 η0,2 η0,2 ) exp(N ) )(− 2 τ
k=0 0 ³s2 −1
=
Y
´ 0 2k exp(−2πiy2 + πi sτ2 n2 − 2πin2 (− sz−µ+s ) exp(N2 )s1 τ
k=0
= exp(−2πis01 (y2 + τν ) − πin2 (− τs )s01 + 2πin2 s01 s τz ) exp(s01 N2 ).
6.2 The functors corresponding to τ 7→ − τ1
79
So far our formula reads: ³ ¡ ¢ ¡ ¢´ Hom γs1 ∗ L(ψ1 ) ⊗ F (− sτ1 N1 ) , γs2 ∗ L(ψ2 ) ⊗ F (− sτ2 N2 ) ∼ =
d−1 M
³ ¡ ¢ ∗ ¡ ¢´ ∗ Hom γ˜1,µ F (V1 , 1, η1 (− zsτ 1 ) exp(N1 )) , γ˜2,µ F (V1 , 1, η2 (− zsτ 2 ) exp(N2 ))
µ=0
∼ =
d−1 M
³ ¡ ¢ Hom F V1 , 1, exp(−2πis02 y1 + πin1 τs s02 + 2πin1 s02 s τz ) exp(s02 N1 ) ,
µ=0
¡ ¢´ F V2 , 1, exp(−2πis01 (y2 + ντ ) − πin2 (− τs )s01 + 2πin2 s01 s τz ) exp(s01 N2 )
And we have the following isomorphisms which then establish the claimed formula: F (V1 , 1, exp(−2πis02 y1 + πin1 τs s02 + 2πin1 s02 s τz + s02 N1 )
(z, v) _
²
²
L(χ1 ) ⊗ F (V1 , − sτ1 N1 )
(z, G1 (z)v)
given by G1 (z) = exp(πi(2s02 y1 )z + πin1 s02 s(− τ1 )z 2 ) exp(−s02 N1 z) and F (V2 , 1, exp(−2πis01 (y2 + τν ) − πin2 (− τs )s01 + 2πin2 s01 s τz + s01 N2 )
(z, v) _
² L(χ2,µ ) ⊗ F (V2 , − sτ2 N2 )
² (z, G2 (z)v)
with G2 (z) = exp(πi(2s01 y2 + n2 µτ )z + πin2 s01 s(− τ1 )z 2 ) exp(−s01 N2 z). It is easily checked that the Gi fulfill the claimed properties. Now we have achieved: ³ ¡ ¢ ¡ ¢´ Hom γs1 ∗ L(ψ1 ) ⊗ F (− sτ1 N1 ) , γs2 ∗ L(ψ2 ) ⊗ F (− sτ2 N2 ) ∼ =
d−1 M
¡ ¢ Hom L(χ1 ) ⊗ F (V1 , − sτ1 N1 ), L(χ2,µ ) ⊗ F (V2 , − sτ2 N2 )
µ=0
∼ =
d−1 M
³ ¡ ∨ τ H 0 L(χ2,µ χ−1 1 ) ⊗ F V2 ⊗ V1 , exp(− s2 N2 +
¢´
τ s1 N1 )
µ=0
and applying νχ
τ τ −1 2,µ χ1 ,− s2 N2 + s1 N1
gives us the claimed expression.
For the third and last step, note that we can arrange our morphism spaces in the
80
Automorphisms of the Elliptic Curve
following diagram ¡ ¢ Hom πs1 ∗ F (η1 N1 ), πs2 ∗ F (η2 N2 )
¡ ¢ / Hom γs1 ∗ F (ψ1 (− τ )N1 ), γs2 ∗ F (ψ2 (− τ )N2 ) s1 s2
∼
² d−1 M
m∗
²
¡ ¢ H 0 L(ξ2,µ ξ1−1 ) ⊗ F (s01 N2 − s02 N1 )
m∗
/
µ=0
d−1 M
∼
¡ τ H 0 L(χ2,µ χ−1 1 ) ⊗ F (− s2 N2 +
¢
τ s1 N1 )
µ=0 ∼
∼
² d−1 M
²
¡ ¢ H 0 L(ξ2,µ ξ1−1 ) ⊗ (V2 ⊗ V1∨ )
m∗
/
µ=0
d−1 M
¡ ¢ ∨ H 0 L(χ2,µ χ−1 1 ) ⊗ (V2 ⊗ V1 )
µ=0
(Here F (M ) stands for a vector bundle with one nontrivial multiplier M ; you always have to fill in the right fibre, in which M is defined; but all spaces were described above in detail, so that no confusion should occur.) The upper square commutes according to lemma 6.11. The lower square is well-known from lemma 6.10 and we can use the formula from this lemma for each µ to get the claimed expression. The next step is to describe Φm∗ =: G, the symplectic counterpart of this functor. As before we look at how m∗ composed with the mirror functors operates on the symplectic categories. Proposition 6.13. Define the functor G = Φm∗ to be the functor that makes the following diagram commutative: Db (E− 1 )
Φ− 1 τ
τ
m∗
/ FK(E − τ1 ) . Φm∗
²
Db (Eτ )
Φτ
² / FK(E τ )
Because Φτ is for each τ an equivalence of categories, each indecomposable object 1 L of FK(E − τ ) is the symplectic counterpart of either a vector bundle or a torsion sheaf. We distinguish several different cases: 1. L is the symplectic counterpart of L(t∗y ζ0 ζ0n−1 ) with y = a(− τ1 ) + b. So L = ³ ¡ ¢´ (Λ, α, M) = (a + t, a(n − 1) + nt), α, C, exp(−2πib) . Then GL = (GΛ, α0 , GM) =
³¡ ¢´ −b + t, −b(n − 1) + nt, α0 , (C, exp(−2πia)) ,
and α0 lies in the same interval as α, and is of course a logarithm of the slope of GΛ. 2. Now suppose L1 , L2 are both like in case 1 symplectic counterparts of line bundles L(t∗yi ζ0 ζ0ni −1 ).Then for the morphism spaces we have M M Hom(L1 , L2 ) = C · em −→ C · ck = Hom(GL1 , GL2 ) em ∈Λ1 ∩Λ2
ck ∈m∗ Λ1 ∩m∗ Λ2
6.2 The functors corresponding to τ 7→ − τ1
em 7→
n−1 X³
81
2
1
(y2 −y1 ) τ τ 2 ) exp(2πi m exp(− πi 4 )( n ) exp(πi n n k)
k=0
´ ¡ 2 2 (b2 −b1 )(a2 −a1 ) ¢ 2 −a1 ) + 4πi exp −πi (b2 −b1 ) −(a c k 2 n n
where n = n2 − n1 . 3. Now suppose L is the symplectic counterpart of a special vector bundle: ¡ ¢ L =Φ− 1 L(t∗y ζ0 ζ0n−1 ) ⊗ F (V, exp N ) ³ τ ¡ ¢´ = (a + t, a(n − 1) + nt), α, V, exp(−2πib + N ) And we get ³ ¡ ¢´ GL = (−b + t, −b(n − 1) + nt), α0 , V, exp(−2πia − τ N ) 4. Now suppose L1 , L2 are both symplectic counterparts of special vector bundles like in item 3. Then G operates on the morphism spaces as 1
Mor(FKE − τ ) −→ Mor(FKE τ ) M M Hom(L1 , L2 ) = V2 ⊗ V1∨ −→ V2 ⊗ V1∨ = Hom(GL1 , GL2 ) em ∈Λ1 ∩Λ2
ck ∈GΛ2 ∩GΛ2
T em
n2 −n X1 −1
7→
K(L1 , L2 )T ck ,
k=0
with 2
2
2 −a1 ) 1 )(a2 −a1 ) K(L1 , L2 ) = exp(−πi (b2 −b1n) 2−(a ) exp(4πi (b2 −b ) −n1 (n2 −n1 )2 ¡ −(b2 −b1 ) ¢ exp n2 −n1 (−τ N2 ⊗ 1V1∨ − 1V2 ⊗ −τ N1∨ ) ¢ ¡ 2 −a1 ) ∨ ∨ exp −(a n2 −n1 (N2 ⊗ 1V1 − 1V2 ⊗ N1 )
5. Now suppose L is the symplectic counterpart of an arbitrary preferred vector bundle, ³ ¡ ¢´ n−1 ∗ L = Φ− 1 πr∗ L(ty η0 η0 ) ⊗ F (V, exp N ) ³τ ¡ ¢´ = pr∗ Φr(− 1 ) L(η) ⊗ F (V, exp N ) τ
then ³ ¡ ¢´ G : L 7→ GL = Φτ γr∗ L(t∗− yτ ψ0 ψ0n−1 ) ⊗ F (V, exp − τr N ) r
Unfortunately, as discussed in detail in chapter 5 of this chapter, we do not know a symplectic counterpart of γ, which serves us as p for π.
82
Automorphisms of the Elliptic Curve 6. Let now denote L1 , L2 two objects of our category, which have arbitrary preferred vector bundles πri ∗ (Ai )as algebraic counterparts: ³ ¡ ¢´ Li = Φ− 1 πri ∗ L(t∗yi η0 η0ni −1 ) ⊗ F (Vi , exp Ni ) τ ³ ¡ ¢´ = pri ∗ Φri (− 1 ) L(η) ⊗ F (Vi , exp Ni ) τ
=: pri ∗ Li Then the involved morphism spaces are Hom(πr1 ∗ A1 , πr2 ∗ A2 ) =
d−1 M
∗ ∗ Hom(˜ π1,µ A1 , π ˜2,µ A2 )
(6.2.3)
∗ ∗ Hom(˜ γ1,µ m∗ A1 , γ˜2,µ m∗ A2 )
(6.2.4)
Hom(˜ p∗1,µ L1 , p˜∗2,µ L2 )
(6.2.5)
Hom(C1 , C2,µ ),
(6.2.6)
µ=0
Hom(m∗ πr1 ∗ A1 , m∗ πr2 ∗ A2 ) =
d−1 M µ=0
Hom(L1 , L2 ) =
d−1 M µ=0
Hom(GL1 , GL2 ) =
d−1 M µ=0
given by 0 0 0 0 (−b1 s2 + t, −b1 s2 (s2 n1 − 1) + s2 n1 t) as underlying Lagrangian C1 = α1 ∈ (− 21 , 12 ) suitable ¡ ¢ M1 = V1 , exp(−2πia1 ) exp(− sτ1 N1 ) 0 0 0 0 (−b2 s1 + t, −b2 s1 (s1 n2 − 1) + s1 n2 t) as underlying Lagrangian 1 1 C2 = α2 ∈ (−³2 , 2 ) suitable ´ ¡ ¢ M2 = V2 , exp −2πi(a2 − n2 µ ) exp(− τ N2 ) s2
s2
Denote by fm,µ the m−th standard basis vector of the µ−th component of this direct sum in formula (6.2.3), i.e. 1 fm,µ = t∗y(µ) θ[ m n , 0](ns(− τ ), nz), n
and consistently em,µ is the m−th intersection point of pullback of L1 and L2 1 to the µ−th copy of E s− τ in formula (6.2.5). Then morphisms are mapped by G according to the formula s01 n2 −s02 n1 −1
T · em,µ 7→
X
³ 1 y2 τ ( s(s0 n2τ−s0 n1 ) ) 2 exp (− πi 4 − πi s0 n2 −s0 n1 ( s2 − 1
k=0
+
2
1
τ (s01 N2 −s02 N1 )2 4πi(s01 n2 −s02 n1 )s
+ πi
n (a2 −a1 − s 2 )2 2 0 0 s1 n2 −s2 n1
− πi
(b1 s02 −b2 s01 )2 s01 n2 −s02 n1
−
2
τ (s01 N2 −s02 N1 ) 0 s(s01 n2 −s02 n1 ) (s1 y2
−
s02 y1
y1 2 s1 )
−
k + 2πi s0 n2 −s 0n m 1 1
2
n2 s1 µ τ )
n a2 −a1 − s 2 µ 2 0 0 s1 n2 −s2 n1
s0 b −s0 b
(s01 N2 − s02 N1 − 2πi s01n22 −s20 n11 ) 1 2 n µ ´ a2 −a1 − s2 b1 s02 −b2 s01 0 0 + s0 n2 −s0 n1 (s1 N2 − s2 N1 − 2πi s0 n2 −s0 n21 ) T · xk,µ −
1
2
1
2
6.2 The functors corresponding to τ 7→ − τ1
83
7. Let now L be ¢the symplectic counterpart of a torsion sheaf, this is L = ¡ Φ− 1 S(y, V, N ) with y = aτ + b. Then τ
1
Ob(FKE − τ ) −→ Ob(FKE τ ) ³ ³ ¡ ¢´ ¡ ¢´ L = (a, t), 21 , V, exp(2πib + N ) 7→ (−b, t), 21 , V, exp(2πia + N )
8. For two objects that are symplectic counterparts of torsion sheaves with different support, the underlying Lagrangians do not intersect and the morphism space is trivial. 9. We still have to view the cases of morphism spaces of objects which are symplectic counterpart of objects A1 , A2 ∈ Db (E− 1 ), where either A1 or A2 are τ torsion sheaves. If A1 is a torsion sheaf, we only have trivial groups on the algebraic and on the corresponding symplectic side. Define ³ ¡ ¢´ L1 := Φ(− 1 ) πr∗ L(t∗y1 ζ0 ζ0n−1 ) ⊗ F (V1 , exp N1 ) τ
L2 := Φ(− 1 ) S(y2 , V2 , N2 ) τ
1 ζ0 = ϕ0 (r(− ), z) τ y1 = a1 r(− τ1 ) + b1 y2 = a2 (− τa ) + b2 Lr Then Hom(L1 , L2 ) ∼ = Hom(Φm∗ L1 , Φm∗ L2 ) and the opera= k=1 V2 ⊗ V1∨ ∼ ∗ tion of Φm on the k−th vectorspace V2 ⊗ V1∨ is given by Φm∗ : T 7→ ³ ´ 2 τ n (a2 +k)2 1 exp −πi nτ b − 2πi b b + πi + 2πi a (a + k) r 2 r 1 2 τ r τ 1 2 ¡ ¢ · exp 4πi(b1 (a2r+k) + a1 b2 + nb2 a2r+k ) ³ · exp −N2 ⊗ (−a1 − n a2r+k − b1 − nb2 )1V1∨ + 1V2 ⊗ (b2 τr −
Proof.
1. By definition of the mirror functor Φ− 1 , τ
L1 = (Λ, α, M) with Λ = (a + t, a(n − 1) + nt) α ∈ (− 12 , 12 ) ¡ ¢ M = C, exp(−2πib)
´
a2 +k ∨ r N1 )
·T
84
Automorphisms of the Elliptic Curve By definition of G ³ ¡ ¢´ G(L1 ) = Φτ m∗ L(t∗y ζ0 ζ0n−1 ) ¢ ¡ = Φτ L(t∗−yτ ϕ0 ϕ0n−1 ) = (Λ0 , α0 , M0 ) with Λ0 = (−b + t, −b(n − 1) + nt) α0 ∈ (− 21 , 12 ) ¡ ¢ M0 = C, exp(−2πia) So G maps ³ ¡ ¢´ (a + t, a(n − 1) + nt), α, C, exp(−2πib) _
³
²
¡ ¢´ (−b + t, −b(n − 1) + nt), α0 , C, exp(−2πia)
2. By definition of G, we determine how it maps morphisms by using the diagram Φ− 1
¡ ¢ Hom L(ζ1 ), L(ζ2 )
τ
/ Hom(L1 , L2 )
m∗
² ¡ ¢Φτ Hom m∗ L(ζ1 ), m∗ L(ζ2 )
G
²
/ Hom(GL1 , GL2 )
And we already know: 1 t∗y θ[ m n , 0](n(− τ ), nz)
Φ− 1 τ
n
1) / exp(−πi (a2 −a1 )2 − 2πi (a2 −a1 )(b2 −b )em n2 −n1 (n2 −n1 )2
m∗
P
²
λk,m (n)t∗− yτ θ[ nk , 0](nτ, nz) n
with
Φτ
²
G
1) / P λk,m (n) exp(−πi (b1 −b2 )2 − 2πi (b1 −b2 )(a2 −a )ck , n2 −n1 (n2 −n1 )2
1
2
y τ τ 2 m λk,m (n) = exp(− πi 4 )( n ) exp(πi n ) exp(2πi n k).
and the sum is running from k = 0 to s01 n2 − s02 n1 . The claim follows by linearity of G. ¢ ¡ 3. Now L = (Λ, α, M) = Φ− 1 L(t∗y ζ0 ζ0n−1 ) ⊗ F (V, exp N ) , with y = a− τ1 + b τ Analogous to the above calculation, applying the definitions yields: ³ ¡ ¢´ GL = Φτ m∗ L(t∗y ζ0 ζ0n−1 ) ⊗ F (V, exp N ) ³ ¡ ¢´ = Φτ L(t∗− yτ ϕ0 ϕn−1 ) ⊗ F V, exp(−τ N ) 0 n ³ ¡ ¢´ = (−b + t, −b(n − 1) + nt), α0 , V, exp(−2πia) exp(−τ N ) .
6.2 The functors corresponding to τ 7→ − τ1
85
4. Let L1 , L2 be both symplectic counterparts of special vector bundles, so ¡ ¢ Li = Φ− 1 L(t∗yi ) ⊗ F (Vi , exp Ni ) , τ
with yi = ai (− τ1 ) + b. Write y = y2 − y1 . Then analogously to the case above we use the definition of G to see that T em is mapped from 2
∗ 1 m 1) y exp(πi (an22−a −n1 )t θ[ n , 0](n(− τ ), nz) ⊗ exp n
7→
n−1 X
exp( −
πi τ 4 )( n )
1 2
k=0
7→
n−1 X
n−1 X
n2 −n1 (N2
¢ 1 − N1∨ − 2πi nb22 −b −n1 ) T
2
∗ m 1 1) y exp(πi nτ y 2 + 2πi nk m) exp(πi (an22−a −n )t θ[ n , 0](n(− τ ), nz) ¢ ¡ a2 −a11 n 1 ⊗ exp n2 −n1 (N2 − N1∨ − 2πi nb22 −b −n1 ) T
¡ 2 2 +b1 ) exp − πi (−b + n2 −n1
¢ πi τ 12 1 + τ N1∨ − 2πi na22 −a −n1 ) exp(− 4 )( n )
−b2 +b1 n2 −n1 (−τ N2
k=0
=
¡ a2 −a1
(a2 − a1 )2 exp(πi nτ y 2 + 2πi nk m) exp(πi ) n2 − n1 ¢ ¡ a2 −a1 1 exp n2 −n1 (N2 − N1∨ − 2πi nb22 −b −n1 ) T ck
K(L1 , L2 )T ck
k=0
5. Follows simply by definition. We cannot calculate a lot, since we know hardly anything about γ and the functor it induces on the symplectic side 6. The final formula is obtained by applying the definitions and former results. Let Ai denote the sheaves Ai = L(ηi ) ⊗ F (Vi , exp Ni ). By definition, T · em,k comes from ¡ (a2 −a1 − n2 )2 fm,k ⊗ exp πi s0 n2 −s0sn21 − 1
2
n
a2 −a1 − s 2 µ 0 2 s01 n2 −s02 n1 (s1 N2
s01 n2 −s02 n1 −1
7→
X
1
+
1 2 τ (s01 N2 −s02 N1 )2 4πi(s01 n2 −s02 n1 )s n (a2 −a1 − s 2 )2 2 s01 n2 −s02 n1
+ πi s01 n2 −s02 n1 −1
X
k=0
1
2
³
y2 τ gk,µ ⊗ ( s(s0 n2τ−s0 n1 ) ) 2 exp − πi 4 − πi s0 n2 −s0 n1 ( s2 −
k=0
7→
s0 b −s0 b ¢ − s02 N1 − 2πi s01n22 −s20 n11 ) T
1
− −
2
τ (s01 N2 −s02 N1 ) 0 s(s01 n2 −s02 n1 ) (s1 y2 n a2 −a1 − s 2 µ 2 s01 n2 −s02 n1
1
2
+
τ (s01 N2 −s02 N1 )2 4πi(s01 n2 −s02 n1 )s n
+ πi
(a2 −a1 − s 2 )2 2 s01 n2 −s02 n1
− πi
(b1 s02 −b2 s01 )2 s01 n2 −s02 n1
−
1
2
n 2 s1 µ τ )
1
y2 τ ( s(s0 n2τ−s0 n1 ) ) 2 exp − πi 4 − πi s0 n2 −s0 n1 ( s2 − 1
−
k + 2πi s0 n2 −s 0n m 1
´ s0 b −s0 b (s01 N2 − s02 N1 − 2πi s01n22 −s02 n11 ) T
³
1
−
s02 y1
y1 2 s1 )
2
τ (s01 N2 −s02 N1 ) 0 s(s01 n2 −s02 n1 ) (s1 y2
y1 2 s1 )
2
k + 2πi s0 n2 −s 0n m 1
− s02 y1 −
1
2
n 2 s1 µ τ )
n
a2 −a1 − s 2 µ 0 2 s01 n2 −s02 n1 (s1 N2
s0 b −s0 b
− s02 N1 − 2πi s01n22 −s02 n11 ) 1 2 n µ ´ a2 −a1 − s2 b1 s02 −b2 s01 0 0 + s0 n2 −s0 n1 (s1 N2 − s2 N1 − 2πi s0 n2 −s0 n21 ) T · xk,µ −
1
2
1
2
The more important question is, whether this is really an element of the morphism space Hom(GL1 , GL2 ). In other words, we still have to show Hom(GL1 , GL2 ) =
d−1 M µ=0
Hom(C1 , Cs,µ ).
86
Automorphisms of the Elliptic Curve But this follows from definition of G. Although we cannot describe the objects GLi , we still have the following diagram, which defines the isomorphism of vectorspaces, we are looking for. Denote by Bi the sheaves ³ ¡ ¢´ Bi = L(ψi ) ⊗ F Vi , exp(− sτi Ni ) , such that
¡ ¢ m∗ πsi ∗ (Ai ) = γsi ∗ (Bi )
with notation like in proposition 6.12. ³ ¡ ¢ ¢ ¡ ¢´ ¡ Hom− τ1 πs1 ∗ (A1 ), πs2 ∗ (A2 ) _ _ _/ Hom ps1 ∗ Φs1 (− 1 ) (A1 ) , ps2 ∗ Φs2 (− 1 ) (A2 ) τ τ ∼
Ld−1 µ=0
∼
²
Φs(− 1 ) τ
∗ ∗ Hom(˜ π1,µ A1 , π ˜2,µ A2 )
²
m∗
¢ ¡ ∗ ∗ (B2 ) ˜1,µ (B1 ), γ˜2,µ µ=0 Hom γ O
Ld−1
Φτ
s
² ¡ ¢ / Ld−1 Hom p˜∗ (ΦA1 ), p˜∗ (ΦA2 ) 1,µ 2,µ µ=0   ÂÂG ®¶ ¡ ¢ / Ld−1 Hom Φ τ γ˜ ∗ (B1 ), Φ τ γ˜ ∗ (B2 ) µ=0 s 1,µ s 2,µ O
∼ Φτ Homτ (γs1 ∗ B1 , γs2 ∗ B2 ) _ _ _ _ _ _ _ _ _/ Homτ (Φτ γs1 ∗ B1 , Φτ γs2 ∗ B2 )
In the lowest square we have an isomorphism on the right hand side going up (dotted arrow) because the diagram commutes. The upper square commutes by definition of Φ and we understand the middle square, which allows us to define then the missing down arrow on the right hand side, which describes G. 7. is a simple application of the definitions. 8. follows from the definitions. 9. On the algebraic side we use the identifications: ¡ ¢ Hom πr∗ (ζ) ⊗ F (V1 , exp N1 ) , S(y2 , V2 , N2 ) ∼ =
² ¡ ¢ ¡ ¢ Hom L(ζ) ⊗ F (V1 , exp N1 ) , πr∗ S(y2 , V2 , N2 ) ∼ =
² ¢ ¡ ¢ ¡ Lr 1 Hom L(ζ) ⊗ F (V , exp N 1 1 ) , S(y2 + k(− τ ), V2 , N2 ) k=1
∼ =
/ Lr
k=1
m∗
² ¢´ ¡ −y2 τ ¢ ¡ Lr τ N ) , S( r + kr , V2 , N2 ) Hom L(ϕ) ⊗ F V , exp(− 1 k=1 r 1 ³
∼ =
² ¡ ¡ ¢ ¢ Hom m∗ πr∗ L(ζ) ⊗ F (V1 , exp N1 ) , m∗ S(y2 , V2 , N2 )
V2 ⊗ V1∨ id
∼ =
² ∨ V k=1 2 ⊗ V1
/ Lr
6.2 The functors corresponding to τ 7→ − τ1
87
with ζ = t∗y1 ζ0 ζ0n−1 ϕ = t∗− y1 τ ϕ0 ϕ0n−1 r
y1 = a1 r(− τ1 ) + b1 y2 = a2 (− τ1 + b2 ). We have the analogous identifications on the symplectic side and can use the k middle square (with strong arrows) to calculate Φm∗ . Denote by Cr(− the 1 ) τ
factor for the k − th component of Φr(− 1 ) and by C kτ the respective factor τ r for the k−th component of Φ τr . Then the k−the component of the action of k −1 and by inserting in the definition of the mirror Φm∗ is given by C kτ (Cr(− 1 ) ) functor we get:
r
τ
¡ ¡ ¢¢ 2 k Cr(− = exp −πir(− τ1 ) n (a2r+k) + 2a1 (a2r+k) 1 2 ) τ ¢ ¡ · exp −2πi( (a2r+k) b1 + a1 b2 + n a2r+k b2 ) ¡ ¢ · exp −N2 ⊗ (a1 + n (a2r+k) )1V1∨ + 1V2 ⊗ (a2r+k) N1∨ ¡ ¡ ¢ ¢ C kτr = exp −πi τr nb22 + 2b1 b2 − 2πi(−b2 a1 − b1 (a2r+k) − nb2 (a2r+k) ) ¡ ¢ · exp −N2 ⊗ (−b1 − nb2 )1V1∨ + 1V2 ⊗ b2 τr N1∨ ¢ ¡ 2 k −1 b22 − 2πi τr b1 b2 + πi nτ (a2 +k) + 2πi τ1 a1 (a2 + k) C kτr (Cr(− = exp −πi nτ 1 ) ) r r τ ¡ ¢ · exp 4πi(b1 (a2r+k) + a1 b2 + nb2 a2r+k ) ¡ ¢ · exp −N2 ⊗ (−a1 − n a2r+k − b1 − nb2 )1V1∨ + 1V2 ⊗ (b2 τr − a2r+k N1∨ )
6.2.2
The pushforward and its symplectic counterpart
Again we consider an explicit isomorphism ¡ ¢ m∗ L(t∗x ϕ0 ϕ0n−1 ) = F C, 1, (t∗−x ϕ0 ϕ−n+1 )(−zτ − τ ) ∼ = L(t∗− x ζ0 ζ0n−1 ), 0 τ
by g(z) = exp(−2πixz + πinτ z 2 ). It is easily checked that since ¡ ¢ g(z + 1) = exp −2πix(z + 1) + πinτ (z 2 + 2z + 1) = exp(−2πix + πinτ + 2πinτ z)g(z) = exp(−2πix − πinτ + 2πinτ z + 2πinτ )g(z) = t∗− x ϕ0 ϕn−1 (−zτ − τ )g(z) 0 n ¡ g(z − τ1 ) = exp −2πix(z − τ1 ) + πinτ (z 2 − 2z τ1 + ¢ ¡ = exp +2πi xτ − 2πinz − πi(− nτ ) g(z) = t∗− x ζ0 ζ0n−1 (z)g(z), n
¢
1 ) τ2
88
Automorphisms of the Elliptic Curve
g really is a holomorphic isomorphism of these bundles. By means of this isomorphism we can calculate how m∗ acts on morphisms of line bundles. First note that for θ[ m n , 0](nτ, nz + x), ¡ ¢ an element of our standard basis of H 0 L(t∗x ϕ0 ϕ0n−1 ) , m m∗ θ[ m n , 0](nτ, nz + x) = θ[ n , 0](nτ, −nzτ + x), ¡ ¢ which is a global section of F C, 1, (t∗−x ϕ0 ϕ−n+1 )(−zτ − τ ) . Using the transforma0 tion formula A.7 we obtain analogously to the calculations for the pullback: 2
m m θ[ m n , 0](nτ, −nzτ + x) = exp(πi n τ − 2πimzτ + 2πi n z)θ(nτ, −nτ z + x + mτ ) 1
2
1 2 m m = exp(− πi 4 )(− nτ ) exp(πi n τ − 2πimzτ + 2πi n x) m 2 1 x n ) nτ )θ(− nτ , z − nτ 2 1 12 x2 exp(− πi 4 )(− nτ ) exp(−πiz nτ − πi nτ + 1 x θ(− nτ , z − nτ −m n)
exp(−πi(z −
=
x nτ
−
−
m n)
2πixz)
This yields θ[ m n , 0](nτ, −nτ z + x)g(z) = 1
1 2 = exp(−2πixz + πinτ z 2 ) exp(− πi 4 )(− nτ ) 2
x 1 exp(−πiz 2 nτ − πi nτ + 2πixz)θ(− nτ ,z − 1
2
x 1 2 1 = exp(− πi 4 )(− nτ ) exp(−πi nτ )θ(− nτ , z −
x m nτ − n ) x m nτ − n )
and, according to formula A.6, 1
n−1 X
1
k=0 n−1 X
1 2 = exp(− πi 4 )(− nτ )
1 2 = exp(− πi 4 )(− nτ )
θ[ nk , 0](n(− τ1 ), nz − ∗ exp(−2πi m n k)t−
k=0
x nτ
x nτ
−
m n)
θ[ nk , 0](n(− τ1 ), nz)
So we have just proven the following lemma. Lemma 6.14. Global sections of L(t∗x ϕ0 ϕn−1 ) are transformed under m∗ into global 0 n−1 ∗ sections of L(t− x ζ0 ζ0 ) by τ
X ¡ x2 ¢ n−1 ¡ ¡ πi ¢¡ 1 ¢ 12 ¢∗ t∗nx θ[ m exp πi exp −2πi m , 0](nτ, nz) → 7 exp − − n 4 nτ nτ n k t− k=0
x nτ
θ
£k
n, 0
¤¡
¢ n(− τ1 ), nz .
1 2
1 For the root (− nτ ) choose the solution that lies in the first quadrant of the complex plane.
Lemma 6.15. Let F1 , F2 be two special vector bundles over Eτ , i.e. Fi = L(t∗xi ϕ0 ϕn0 i −1 ) ⊗ F (Vi , exp Ni ). ¡ ¢ Hom(F1 , F2 ) ∼ = H 0 L(ϕ) ⊗ V with ϕ = t∗x2 −x1 ϕ0 ϕ0n2 −n1 −1 and V = V2 ⊗ V1∨ . With this notation
6.2 The functors corresponding to τ 7→ − τ1
89
¡ ¢ n −1 1. m∗ Fi ∼ = L(t∗− xi ζ0 ζ0 i ) ⊗ F Vi , exp(− Nτ ) τ
2. And if n2 > n1 , morphisms are mapped as ¡ ¢ ¡ ¢ H 0 L(ϕ) ⊗ V −→ H 0 L(ζ) ⊗ V n−1 X ¤ ¡ ¢¡ 1 ¢ 12 ¡ x2 ¢ t nx θ n , 0 (nτ, nz) ⊗ T → 7 exp − πi − nτ exp πi nτ + 2πi nk m 4 ¢ £ k ¤¡ n k=0 N2 x θ t∗− nτ n , 0 − τ , nz ⊗ exp(− 4πinτ − ∗
Proof.
£m
Nx nτ )T
1. m∗ (Fi ) = m∗ F (Vi , ϕi exp Ni ). A calculation on the fibres shows ¡ ¢ m∗ F (Vi , ϕi exp Ni ) = F Vi , 1, ϕi (−zτ − τ )−1 exp(−N ) .
And then we use isomorphism of holomorphic vector bundles ¡ ¢ ¡ ¢ G : F Vi , 1, ϕi (−zτ − τ )−1 exp(−N ) − → L(ζi ) ⊗ F Vi , exp(− Nτ i ) (z, v) 7→ (z, G(z)v) with G(z) = exp(−2πixz + πinτ z 2 + zN ). 2. Consider the following diagram ¡ ¢ H 0 F (V, ϕ exp N ) o ²
νϕ,N
¡ ¢ H 0 L(ϕ) ⊗ V Â Â
m∗
Â
³ ¡ ¢´ H 0 F V, 1, ϕ(−zτ − τ )−1 exp(−N ) Â Â Â Â
G
² ³ ¡ ¢´ H 0 F V, ζ exp(− Nτ ) o
Â
ν
N ζ,− τ
H0
¡ ² ¢ L(ζ) ⊗ V
We are interested in the dotted arrow. By using its commutativity we obtain: t∗x θ[ m n , 0](nτ, nz) ⊗ T n
is mapped by νϕ,N to N d ∗ m m exp(− 2πin dz )t x θ[ n , 0](nτ, nz) · T = θ[ n , 0](nτ, nz + x − n
N 2πi )
·T
applying m∗ yields θ[ m n , 0](nτ, −nzτ + x −
N 2πi )
·T
applying G yields N exp(−2πixz + πinτ z 2 + zN )θ[ m n , 0](nτ, −nzτ + x − 2πi ) · T ¡ 2 = exp −2πixz + πinτ z 2 + zN + πi mn τ + 2πi m n (−nτ z + x −
θ(nτ, −nzτ + x
¢
N 2πi ) N − 2πi
+ mτ ) · T
90
Automorphisms of the Elliptic Curve now we use the formula from A.8 ¡ 2 = exp − 2πixz + πinτ z 2 + zN + πi mn τ + 2πi m n (−nτ z + x −
¢
1
N 2πi ) 2 x N exp(−πinτ (−z + nτ − 2πinτ +m n) ) 1 x N θ(− nτ , −z + nτ − 2πinτ +m n)·T
1 2 exp(− πi 4 )(− nτ )
πi x2 1 12 N2 1 x N )(− nτ ) exp(−πi nτ − 4πinτ )θ(− nτ , −z + nτ − 2πinτ +m n)·T 4 πi x2 1 12 1 x N = exp(− )(− nτ ) exp(−πi nτ )θ(− nτ , +z − nτ + 2πinτ −m n) 4 N2 exp(− 4πinτ )·T = exp(−
−1 by applying νζ,− N we get τ
πi x2 1 12 1 x )(− nτ ) exp(−πi nτ )θ(− nτ , +z − nτ − 4 n−1 X 1 12 k x2 exp(− πi = 4 )(− nτ ) exp(πi nτ + 2πi n m)
m n)
exp(−
2
N ⊗ exp(− 4πinτ )T
k=0
t∗−
2
x nτ
N θ[ nk , 0](− nτ , nz) ⊗ exp(− 4πinτ −
Nx nτ )
·T
Proposition 6.16. An indecomposable vector bundle is isomorphic to ¡ ¢ ) ⊗ F (V, exp N ) . πr∗ Lrτ (t∗x ϕ0 ϕn−1 0 m∗ operates on this objects by ¡ ¢ Ob Db (Eτ ) −→ Ob(Db E− 1 ) τ ¡ ¢ n−1 ∗ ∗ N πr∗ Lτ (tx ϕ0 · ϕ0 ) ⊗ F (V, exp N ) 7→ γr∗ L− 1 (t− x ζ0 · ζ0n−1 ) ⊗ F (V, exp − rτ ) , rτ
τ
where γr∗ is the functor in proposition 5.6. Given two vector bundles πsi ∗ Ai , the morphism spaces are Hom(πs1 ∗ (A1 ), πs2 ∗ (A2 ))
m∗
¡
¢
/ Hom γs ∗ (m∗ A1 ), γs ∗ (m∗ A2 ) 1 2 ∼
∼
Ld−1
µ=0 H
0
²¢
¡ L(ζ) ⊗ (V2 ⊗ V1∨ ) /
Ld−1
µ=0 H
0
²¢
¡ L(ξ) ⊗ (V2 ⊗ V1∨ )
and morphisms are mapped by m∗ according to the formula 1
1 2 fm,µ ⊗ T 7→ exp(− πi 4 )(− τ sn )
n−1 X
2
k exp(πi x(µ) nsτ + 2πi n m)gk,µ
k=0
⊗ exp(−
(s01 N2 −s02 N1 )2 4πinsτ
−
(s01 N2 −s02 N1 )x(µ) )T nsτ
6.2 The functors corresponding to τ 7→ − τ1
91
with the abbreviations d = gcd(s1 , s2 ) si s0i = d s1 s2 s= d n = n2 − n1 x(µ) = s01 x2 − s02 x1 − n2 s01 µτ ψ0,i = ϕ0 (si τ ) ni −1 ψi = t∗xi ψ0,i ψ0,i
Ai = L(ψi ) ⊗ F (Vi , exp Ni ) ζ0 = ϕ0 (sτ ) s 0 n1
ζ1 = t∗s0 x1 ζ0 ζ0 2 2
ζ2,µ = ζ= ξ0 = ξ=
s0 n −1 t∗s0 x2 +n2 s0 µτ ζ0 ζ0 1 2 1 1 ζ2,µ ζ1−1 1 ) ϕ0 (− sτ s0 n −s0 n t∗ x(µ) ξ0 ξ02 1 1 2 − sτ
and finally the bases of our spaces of morphisms are fm,µ = t∗x(µ) θ[ m n , 0](nsτ, nz) n
gk,µ =
1 t∗ x(µ) θ[ nk , 0](n(− sτ ), nz) − nsτ
Proof. We use the commutative diagram Erτ
m
²
1 rτ
r γ
πr
Eτ
/ E−
m
² / E− 1 τ
to get the identity ³ ¡ ³ ¡ ¢´ ¢´ ) ⊗ F (V, exp N ) m∗ πr∗ Lτ (t∗x ϕ0 · ϕn−1 ) ⊗ F (V, exp N ) = γr∗ m∗ Lτ (t∗x ϕ0 · ϕn−1 0 0 ¡ ¢ n−1 N x ζ · ζ = γr∗ L− τ1 (t∗− rτ ) ⊗ F (V, exp − rτ ) . 0 0
and then we use lemma 6.15 to see that this is isomorphic to the bundle in the proposition. For the second part of the proposition we use again the commutativity of the diagram in lemma 6.11 and can then use the formula for special bundles of lemma 6.10. Proposition 6.17. Define the functor G := Φm∗ to be the functor that makes the
92
Automorphisms of the Elliptic Curve
following diagram commutative: Db (Eτ ) m∗
²
Db (E− 1 ) τ
Φτ
Φ− 1 τ
/ FK(E τ ) . ²
Φm∗
/ FK(E − τ1 )
An indecomposable object of FK(E τ ) is the symplectic counterpart of a coherent sheaf. So it suffices to consider all the cases of proposition 6.16 1. Let L be the symplectic counterpart of a line bundle. Then ³ ¡ ¢ ¡ ¢´ L = Φτ Lτ (t∗x ϕ0 · ϕn−1 ) = (a + t, a(n − 1) + nt), α, C, exp(−2πib) 0 ³ ¡ ¢´ L 7→ (b + t, b(n − 1) + nt), α0 , C, exp(2πia) ¡ ¢ = Φ− 1 L− 1 (t∗− x ζ0 · ζ0n−1 ) τ
τ
τ
2. If L1 , L2 are both symplectic counterparts of line bundles, we get for their morphisms (use x = x2 − x1 , n = n2 − n1 ): M M Hom(L1 , L2 ) = C · em −→ C · ek = Hom(GL1 , GL2 ) em ∈Λ1 ∩Λ2
ek ∈GΛ1 ∩GΛ2
by em 7→
n−1 X
1
2
1 2 x m (− nτ ) exp(− πi 4 + πi nτ − 2πi n k) ¡ k=0 2 2 −a1 )(b2 −b1 ) ¢ 2 −a1 ) exp −πi (b2 −b1n) 2−(a + 4πi (a2(n ek 2 −n1 2 −n1 )
3. Now consider an object that is the symplectic counterpart of a special vector bundle. ¡ ¢ ) ⊗ F (V, exp N ) L :=Φτ Lτ (t∗x ϕ0 · ϕn−1 0 ³ ¡ ¢´ = (a + t, a(n − 1) + nt), α, V, exp(−2πib + N ) L is mapped to Φ− 1 L− 1 (t∗− x ζ0 · ζ0n−1 ) ⊗ F (V, exp − Nτ ) τ τ ³ τ ¡ = (b + t, b(n − 1) + nt), α0 , V, exp(2πia −
¢´
N τ )
.
4. For two objects L1 , L2 that are symplectic counterparts of special vector bundles we get for the morphisms between them M M Hom(L1 , L2 ) = V2 ⊗ V1∨ · em −→ V2 ⊗ V1∨ · ek = Hom(GL1 , GL2 ) em ∈Λ1 ∩Λ2
ek ∈GΛ1 ∩GΛ2
6.2 The functors corresponding to τ 7→ − τ1
93
by means of T · em 7→
n−1 X
2
2
2 −a1 ) 1 )(b2 −b1 ) exp(−πi (b2 −b1n) 2−(a + 4πi (a2 −a ) −n1 (n2 −n21 ) ¡ ¢ k=0 N1 1 N2 ∨ exp nb22 −b −n1 ( τ ⊗ 1V1 − 1V2 ⊗ τ ) ¡ ¢ −a1 exp − na22 −n (N2 ⊗ 1V1∨ − 1V2 ⊗ N1∨ ) 1 1
2
(x2 −x1 ) k 1 2 exp(− πpi 4 )(− (n2 −n1 )τ ) exp(πi (n2 −n1 )τ + 2πi n2 −n1 m)T · ek
5. Now suppose L is the symplectic counterpart of an arbitrary preferred vector bundle, ³ ¡ ¢´ L = Φτ πr∗ L(t∗y η0 η0n−1 ) ⊗ F (V, exp N ) ³ ¡ ¢´ = pr∗ Φrτ L(η) ⊗ F (V, exp N ) then ³ ¡ ¢´ N G : L 7→ GL = Φ− 1 γr∗ L(t∗− yτ ψ0 ψ0n−1 ) ⊗ F (V, exp − rτ ) τ
r
Unfortunately, as discussed in detail in chapter 5 of this chapter, we do not know a symplectic analogue of γ. 6. Let now denote L1 , L2 two objects of our category, which have arbitrary preferred vector bundles πri ∗ (Ai ) as algebraic counterparts: ³ ¢´ ¡ Li = Φτ πri ∗ L(t∗xi ψ0 ψ0ni −1 ) ⊗ F (Vi , exp Ni ) ³ ¡ ¢´ = pri ∗ Φri τ L(ψi ) ⊗ F (Vi , exp Ni ) =: pri ∗ Li Then the involved morphism spaces are Hom(πr1 ∗ A1 , πr2 ∗ A2 ) =
d−1 M
∗ ∗ Hom(˜ π1,µ A1 , π ˜2,µ A2 )
(6.2.7)
µ=0
Hom(m∗ πr1 ∗ A1 , m∗ πr2 ∗ A2 ) =
d−1 M
∗ ∗ Hom(˜ γ1,µ m∗ A1 , γ˜2,µ m∗ A2 )
µ=0
Hom(L1 , L2 ) =
d−1 M
Hom(˜ p∗1,µ L1 , p˜∗2,µ L2 )
µ=0
Hom(GL1 , GL2 ) =
d−1 M µ=0
Hom(C1 , C2,µ ),
(6.2.8)
94
Automorphisms of the Elliptic Curve given by 0 0 0 0 (b1 r2 + t, b1 r2 (r2 n1 − 1) + r2 n1 t) as underlying Lagrangian C1 = α1 ∈ (− 12 , 12 ) suitable ¡ r0 N ¢ M1 = V1 , exp(2πia1 ) exp(− 2rτ 1 ) 0 0 0 0 (b2 r1 + t, b2 r1 (r1 n2 − 1) + r1 n2 t) as underlying Lagrangian 1 1 C2 = α2 ∈ (−³2 , 2 ) suitable ´ ¡ ¢ 0 M2 = V2 , exp 2πi(a2 − n2 µ ) exp(− r1 N2 ) r2
rτ
Denote by fm,µ the m−th standard basis vector of the µ−th component of this direct sum in formula (6.2.7), i.e. fm,µ = t∗x(µ) θ[ m n , 0](nrτ, nz), n
and consistently em,µ is the m−th intersection point of pullback of L1 and L2 to the µ−th copy of E rτ in formula (6.2.8). Then morphisms are mapped by G according to the formula T · em,µ 7→
R X k=0
³ (a + n2 µ −a )2 ´ 1 2 1 x(µ)2 r2 1 k 2 exp πi( ) exp(− πi n2 −n1 4 )(− τ rn ) exp(πi nrτ + 2πi n m) ³ a + n2 µ −a ¡ ´ 2 1 r 0 b2 −r10 b1 ¢ ∨ 0 0 2 ∨ 2 ∨ − 1V ⊗ r N − 2πi 1 exp − n2r−n N ⊗ 1 r V ⊗V V 1 2 2 1 2 n2 −n1 2 1 1 1 n2 µ ³ 0 ´ 0 ¡ −a2 + r +a1 ¢ ∨ r b −r b 2 exp 1n22 −n21 1 − rN22τ ⊗ 1V1∨ + 1V2 ⊗ rN11τ − 2πi1V2 ⊗V1∨ n2 −n1 ³ ´ (r 0 N2 −r20 N1 )2 (r 0 N −r20 N1 )x(µ) (r 0 b −r 0 b )2 exp(− 1 4πinrτ − 1 2 nrτ ) exp −πi( 1 n22 −n21 1 ) T · xk,µ ,
with R = r10 n2 − r20 n1 − 1. ¡ ¢ 7. Let now L = Φτ S(x, V, N ) be the symplectic counterpart of a torsion sheaf, with x = aτ + b. Then 1
Ob(FKE τ ) −→ Ob(FKE − τ ) ³ ³ ¡ ¢´ ¡ ¢´ L = (a, t), 12 , V, exp(2πib + N ) 7→ (b, t), 12 , V, exp(−2πia + N )
8. For two objects that are symplectic counterparts of torsion sheaves with different support, the underlying Lagrangians do not intersect and the morphism space is trivial. 9. We still have to view the cases of morphism spaces of objects which are symplectic counterpart of objects A1 , A2 ∈ Db (E− 1 ), where either A1 or A2 is τ a torsion sheaf. If A1 is a torsion sheaf, we only have trivial groups on the algebraic and on the corresponding symplectic side. If A1 is a preferred vector
6.2 The functors corresponding to τ 7→ − τ1
95
bundle, and A2 is a torsion sheaf, with ¡ ¢ A1 = πr∗ L(t∗x1 ψ0 ψ0n−1 ) ⊗ F (V1 , exp N1 ) A2 = S(x2 , V2 , N2 ) x1 = a1 rτ + b1 x2 = a2 τ + b2 Li = Φτ Ai L ∨ then Hom(L1 , L2 ) = k=1 1rV2 ⊗ V1 = Hom(Φm∗ L1 , Φm∗ L2 ). And on the k−th component Φm∗ operates as ¡ ¢ 2 1 Φm∗ : T 7→ exp πirτ (n (a2r+k) + 2a1 (a2r+k) ) + πi rτ (nb22 + 2b1 b2 ) 2 ¡ ¢ · exp 4πi(b1 a2r+k + a1 b2 + nb2 a2r+k ) ¡ ¢ b2 · exp −N2 ⊗ (b1 + nb2 − a1 + n a2r+k )1V1∨ − 1V2 ⊗ ( rτ + a2r+k )N1∨ T Proof. The formulas are all obtained by applying the definition of Φm∗ , calculations are not involved since the functor m∗ scrambles the data of the involved bundles in such a way that for the correction terms of Φm∗ , no cancelations occur. Denote by C(τ, A) the correction term for morphisms involved in the functor Φτ : Hom(A1 , A2 ) − → Hom(L1 , L2 ), by C(− τ1 , m∗ A) the corresponding correction → term for Φ− 1 and by Dk (A) the correction terms for the functor Hom(A1 , A2 ) − τ Hom(m∗ A1 , m∗ A2 ). Then T e ∈ Hom(L1 , L2 ) is mapped by G to X C(τ, A)−1 Dk (A)C(− τ1 , m∗ A)T ek . k
For the different cases, we now only have to choose the right correction terms, according to the previous sections. 1
Between E τ and E − τ there is neither a complex nor real symplectic map because in general τ and − τ1 are not integer multiples of each other. Also Im τ = y y and Im − τ1 = x2 +y 2 (x = Re τ ) are not in integer relation to each other, and so the functors Φm∗ or Φm∗ are for trivial reason not induced by morphisms of the underlying manifolds. This means, the functor Φ on the morphisms in the category pDb ell, does not take its images in the category pFKell. To make it a functor, we need to extend the category and allow more functors. So would it suffice, to allow functors induced by correspondences - whatever notion of correspondence we would want to use? Unfortunately the answer is no. Under a correspondence two objects in FKE τ with the same underlying Lagrangian, for which the local systems have the same rank but differ in the monodromy, are mapped to objects with again the same underlying Lagrangian; independently on how this correspondence might look like. But the objects have different underlying Lagrangians under the functor Φm∗ . The same is true for the pushforward. So m∗ and m∗ in pDb ell do not have symplectic counterparts that we can understand geometrically. The following theorem summarizes the results of this section: Theorem 6.18. The mirror functor depends on the choice of a lattice Λτ for the elliptic curve. Two different choices differ by an equivalence that we cannot interpret (geometrically) on the symplectic side.
96
Automorphisms of the Elliptic Curve
Chapter 7
Functors Induced by Symplectic Morphisms In the previous chapters, we always started on the algebraic side and looked at pushforwards and pullbacks along morphisms of elliptic curves. Now we have a look at the symplectic category and the functors induced by symplectic morphisms on the algebraic side. Before we can engage in such a study, we first have to define what a morphism should be. We have not yet decided whether we want to consider real or complex symplectic morphisms, by which we denote maps respecting the imaginary part or the whole complexified symplectic structure, respectively. The B-field of E τ is only determined up to Z, so we would like to choose a notion of morphism, such that E τ and E τ +1 are isomorphic. Polishchuk and Zaslow [63] for comparison, usually write E q with q = exp(2πiτ ) instead of E τ , to emphasis this. If we consider complex symplectic maps there are no morphisms at all between E τ and E τ +1 because τ + 1 is not an integer multiple of τ , much less an isomorphism. If we 0 define Hom(E τ , E τ ) as the set of real symplectic functions that means, functions 0 f : Eτ − → E τ which fulfill f ∗ (adx ∧ dy) = a0 dx ∧ dy for τ = ai + b and τ 0 = a0 i + b0 , then the identity is a map from E τ − → E τ +1 . The identity is of course an isomorphisms; so this definition looks preferable. Now we have a look at the corresponding Fukaya categories. Of course an isomorphism should induce an equivalence of the categories, but: Theorem 7.1. The real symplectic identity E τ −→ E τ +1 does not induce a functor FK(E τ ) −→ FK(E τ +1 ). We will use the mirror functor to prove the theorem: Proof. The identity induces maps on the objects and morphisms given by FK(E τ ) 3 (Λ, α, M) 7→ (Λ, α, M) ∈ FK(E τ +1 ) and f ∈ Homτ (L1 , L2 ) 7→ f ∈ Homτ +1 (L1 , L2 ).
98
Functors Induced by Symplectic Morphisms
For this to be functorial, the composition has to be the same in both categories: (f ◦τ g) = (f ◦τ +1 g). Let us suppose that f = u·x1 and g = v·x2 are both concentrated at the intersection points x1 , x2 respectively. Then we get at the intersection point x3 : (v·x2 ◦τ u·x1 )x3 · x3 =
X
Z φ∗ τ dx ∧ dy)P (M3 ) ◦ v ◦ P (M2 ) ◦ u ◦ P (M1 ) · x3
exp(2πi
φ
=
X
exp(2πiτ Aφ )P (M3 ) ◦ v ◦ P (M2 ) ◦ w ◦ P (M1 ) · x3
φ
(v·x2 ◦τ +1 u ·x1 )x3 · x3 =
X
Z exp(2πi
φ∗ (τ + 1)dx ∧ dy)P (M3 )◦v◦P (M2 )◦u◦P (M1 )·x3
φ
=
X
exp(2πi(τ + 1)Aφ )P (M3 ) ◦ v ◦ P (M2 ) ◦ u ◦ P (M1 ) · x3
φ
=
X
exp(2πiAφ ) exp(2πiτ Aφ )P (M3 ) ◦ v ◦ P (M2 ) ◦ u ◦ P (M1 ) · x3 .
φ
And since in general Aφ ∈ / Z, it is not immediate that this is equal in general. Now suppose Li are all symplectic counterparts of special vectorbundles, so ¡ ¢ Li = Φτ L(t∗xi ϕ0 ϕn0 i −1 ) ⊗ F (Vi , exp Ni ) = Φτ (Ai ). Then the algebraic counterparts (with respect to Φτ ) of f = u · ek and g = v · dl are: ¢ £ k ¤¡ C(A1 , A2 )−1 · u ⊗ t∗x2 −x1 θ n2 −n (n − n )τ, (n − n )z 2 1 2 1 1 n2 −n1
= C(A1 , A2 )−1 · u ⊗ fk,τ and C(A2 , A3 )−1 · v ⊗ t∗x3 −x2 θ n3 −n2
£
l n3 −n2
¤¡
(n3 − n2 )τ, (n3 − n2 )z
¢
= C(A2 , A3 )−1 · v ⊗ gl,τ with ¡ ¡ ¢ 2¢ ∨ b2 −b1 1) 1 C(A1 , A2 )−1 = exp πi (an22−a exp − na22 −a −n1 −n1 (N2 ⊗ 1 − 1 ⊗ N1 − 2πi1 n2 −n1 ) ¡ ¡ ¢ 2¢ ∨ b3 −b2 2) 2 C(A2 , A3 )−1 = exp πi (an33−a exp − na33 −a −n2 −n2 (N3 ⊗ 1 − 1 ⊗ N2 − 2πi1 n3 −n2 ) The composition is according to proposition 2.15 given by: ¡
¢ ¡ ¢ C(A1 , A2 )−1 · u ⊗ fk,τ ◦τ C(A2 , A3 )−1 · v ⊗ gl,τ ¡ ¢ n2 −N1 −n1 N2 n1 N2 −n2 N1 )gl,τ (z + 2πin ) C(A1 , A2 )−1 · u ⊗ C(A2 , A3 )−1 · v . (7.0.1) =fk,l (z − 2πin 1 (n1 +n2 ) 2 (n1 −n2 )
The algebraic counterparts of the Li under Φτ +1 are with xi = ai τ + bi as above, such that ai (τ + 1) + bi = xi + ai : L(t∗xi +ai ψ0 ψ0ni −1 ) ⊗ F (Vi , exp Ni ) =: A˜i .
99 The algebraic counterparts of the morphisms f and g und Φτ +1 are therefore: C(A˜1 , A˜2 )−1 · u ⊗ t∗x2 −x1 +a2 −a1 θ
£
n2 −n1
k n2 −n1
¤¡
(n2 − n1 )(τ + 1), (n2 − n1 )z
¢
= C(A˜1 , A˜2 )−1 · u ⊗ fk,τ +1 and C(A˜2 , A˜3 )−1 · v ⊗ t∗x3 −x2 +a3 −a2 θ n3 −n2
£
l n3 −n2
¤¡
(n3 − n2 )(τ + 1), (n3 − n2 )z
¢
= C(A˜2 , A˜3 )−1 · v ⊗ gl,τ +1
with ¡ ¢ ¡ 2¢ ∨ b2 −b1 1) 1 C(A˜1 , A˜2 )−1 = exp πi (an22−a exp − na22 −a −n1 −n1 (N2 ⊗ 1 − 1 ⊗ N1 − 2πi1 n2 −n1 ) C(A˜2 , A˜3 )−1
= C(A1 , A2 )−1 ¡ ¡ ¢ 2¢ ∨ b3 −b2 2) 2 = exp πi (an33−a exp − na33 −a −n2 −n2 (N3 ⊗ 1 − 1 ⊗ N2 − 2πi1 n3 −n2 ) = C(A2 , A3 )−1
For functoriality the composition with respect to τ + 1 has to give the same as the composition with respect to τ. According to proposition 2.15 we have: ¢ ¢ ¡ ¡ C(A˜1 , A˜2 )−1 · v ⊗ fk,τ +1 ◦τ +1 C(A˜2 , A˜3 )−1 · v ⊗ gl,τ +1 ¡ ¢ n1 N2 −n2 N1 n2 −N1 −n1 N2 ˜1 , A˜2 )−1 · v ⊗ C(A˜2 , A˜3 )−1 · v =fk,τ +1 (z − 2πin g (z + ) C( A l,τ +1 2πin2 (n1 −n2 ) 1 (n1 +n2 ) ¡ ¢ −1 −1 n1 N2 −n2 N1 2 −N1 −n1 N2 =fk,τ +1 (z − 2πin1 (n1 +n2 ) )gl,τ +1 (z + n2πin ) C(A , A ) · v ⊗ C(A , A ) · v 1 2 2 3 2 (n1 −n2 ) !
=formula (7.0.1), so we have to show that fk,τ +1 (z)gl,τ +1 (z) = fk,τ (z)gl,τ (z). And we see: fk,τ +1 (z)gl,τ +1 (z)
£ k ¤¡ ¢ = t∗x2 −x1 +a2 −a1 θ n2 −n (n2 − n1 )(τ + 1), (n2 − n1 )z 1 n2 −n1 £ l ¤¡ ¢ t∗x3 −x2 θ n3 −n (n3 − n2 )τ, (n3 − n2 )z 2 n3 −n2 £ k ¤¡ ¢ ∗ = t x2 −x1 θ n2 −n (n2 − n1 )τ, (n2 − n1 )z + a2 − a1 1 n2 −n1 £ l ¤¡ ¢ t∗x3 −x2 θ n3 −n (n3 − n2 )τ, (n3 − n2 )z + a3 − a2 2 n3 −n2
= fk,τ (z +
a2 −a1 n2 −n1 )gl,τ (z
+
a3 −a2 n3 −n2 )
100
Functors Induced by Symplectic Morphisms
and this yields according to the Riemann formula A.5 X ¡ 1 )d+k+l λd (p(τ + 1))θ[ (n2 −n , 0] (n3 − n1 )τ, (n3 − n1 )(z + = n3 −n1
¢
x3 −x1 +a3 −a1 ) n3 −n1
Z d∈ (n −n )Z 3 1
X
=
λd (p(τ + 1))hm(d),τ +1 (z),
Z d∈ (n −n )Z 3 1
with ¢ )(n3 −n2 )d+(n3 −n2 )k−(n2 −n1 )l ¡ ] (n2 − n1 )(n3 − n2 )(n3 − n1 )τ, x , λd (x) = θ[ (n2 −n1 (n 2 −n1 )(n3 −n2 )(n3 −n1 ) 1 +a2 −a1 p(τ + 1) = (n2 − n1 )(n3 − n2 )( x2 −x − n2 −n1
and m(d) =
(n2 −n1 )d+k+l n3 −n1
x3 −x2 +a3 −a2 ) n3 −n2
and Φτ +1 (hk,τ +1 ) = dk ∈ {Λ1 ∩ Λ3 }.
While fk,τ (z)gl,τ (z) X =
¡ 1 )d+k+l λd (p(τ ))θ[ (n2 −n , 0] (n3 − n1 )τ, (n3 − n1 )(z + n3 −n1
x3 −x1 n3 −n1 )
¢
Z d∈ (n −n )Z 3 1
=
X
λd (p(τ ))hm(d),τ ,
Z d∈ (n −n )Z 3 1
with m(d) like above, Φτ (hk,τ ) = dk ∈ {Λ1 ∩ Λ3 }, and +a2 −a1 p(τ + 1) = (n2 − n1 )(n3 − n2 )( x2 −xn21−n − 1 1 = (n2 − n1 )(n3 − n2 )( nx22 −x −n1 −
= p(τ ) + (n2 − n1 )(n3 −
x3 −x2 +a3 −a2 ) n3 −n2
x3 −x2 n3 −n2 ) + (n2 − a3 −a2 1 n2 )( na22 −a −n1 − n3 −n2 ).
−a1 n1 )(n3 − n2 )( na22 −n − 1
a3 −a2 n3 −n2 )
For this to be a functor, we need therefore λd (p(τ )) = λd (p(τ + 1)) (these define the coefficients of the basis vectors) and this implies p(τ ) has to equal p(τ + 1) modulo Z. But −a1 2 − na33 −a /Z (n2 − n1 )(n3 − n2 )( na22 −n −n2 ) ∈ 1 in general. Remark 7.2. Of course FK(E τ ) and FK(E τ +1 ) are equivalent categories. The equivalence is induced by the algebraic basechange τ 7→ τ + 1. In section 6.1 we explicitly described this equivalence also between the Fukaya categories. The remarkable point here is that, without previous knowledge of a mirror symmetry we had to choose a lift of the B− field to define our category and it then depends on this choice in a noncanonical and not intrinsically interpretable way. Theorem 7.3. The definition of the Fukaya category of E q , with q = exp(2πiτ ) depends on the choice of a lift of the B-field (that is on the specific choice of such a τ ). The categories constructed for two different choices differ by an equivalence that has no (geometric) interpretation.
101 Proof. Since the natural map between the manifolds for different choices, the identity, does not induce a functor and much less an equivalence (see theorem 7.1), we are left only with the symplectic counterpart of the algebraic base change, for which we have no (geometric) interpretation (see section 6.1). A corollary of theorem 7.1 is the following Theorem 7.4. Real symplectic morphisms between complexified symplectic tori induce in general not functors between the respective Fukaya categories. Therefore this is not a good notion of morphism for us, since we are mainly interested in the induced functors. Complex symplectic maps, however, do the job for us. Remark 7.5. We only want to consider affine symplectic maps, because we are interested in the functors they induce. For a special Lagrangian Λ its image under f is always special if and only if f is affine. 0
Definition and Remark 7.6. We define the elements in Hom(E τ , E τ) in the category of complexified symplectic tori to be the linear complex symplectic maps 0 f : Eτ − → E τ . f being linear complex symplectic means there is a matrix µ ¶ a b A= ∈ Mat2 (Z) c d such that f (x, y) = A(x, y) = (ax + by, cx + dy). The entries of A have to be integer, so that f is welldefined on the torus. Since τ 0 = det(A)τ, and τ, τ 0 ∈ H, the determinant det A is always positive. Proof. Since f is a symplectic map, the following equation is fulfilled for any two vectors v, w ∈ Tx E τ at a point x: ¡ 0 ¢ ¡ ¢¡ ¢ τ dx ∧ dy (v, w) = τ dx ∧ dy f (v), f (w) . This yields: ¯ ¯ ¯ ¯ ¯av1 + bv2 aw1 + bw2 ¯ ¯ ¯ 0 ¯v1 w1 ¯ ¯ ¯ =τ¯ τ ¯ cv1 + dv2 cw1 + dw2 ¯ v2 w2 ¯ ³ ´ = τ (av1 + bv2 )(cw1 + dw2 ) − (aw1 + bw2 )(cv1 + dv2 ) ³ ´ = τ (ac − ac)v1 w1 − (bd − bd)v2 w2 + (ad − bc)v1 w2 − (ad − bc)v2 w2 ¯ ¯ ¯v w1 ¯ ¯. = τ det(A) ¯¯ 1 v2 w2 ¯
102
Functors Induced by Symplectic Morphisms 0
Now we want to define for a morphism f : E τ − → E τ the pushforward and pullback on the Fukaya categories. But not morphisms as such, graded morphisms operate naturally on the Fukaya category. A graded morphism is a pair (f, γ), where f is a complexified linear symplectic map and γ ∈ g + Z, where πig is the amount by which f rotates the imaginary axis. Composition is defined by (f, γ) ◦ (f 0 , γ 0 ) = (f ◦ f 0 , γ + γ 0 ). For a more detailed discussion of this (though also more general) see the book of Seidel [67] chapter II, sections 10 and 14. We define the pushforward as follows: 0
(f, γ)∗ : Ob(FKE τ ) − → Ob(FKE τ ) ¡ ¢ ¡ ¢ Λ, α, M 7→ f (Λ), f∗ α, (f |Λ )∗ (M) , where ¡ (f |Λ )∗ M ¢ is the direct image of the sheaf M. f∗ α is defined as follows: for Λ = x(t), y(t) , α ∈ (− 12 + k, 12 + k)¡⊂ R, k ∈ Z ¢ is defined such that x(0) + iy(0) + exp(πiα) ∈ Λ. Define fα such that f exp(πiα) = exp(πif∗ α); where we choose for f∗ α that element in R, such that it f∗ α ∈ (− 21 + k + γ, 21 + k + γ). 0
(f, γ)∗ : Mor(FKE τ ) −→ Mor(FKE τ ) ³ ´ ³ ´ Hom (Λ1 , α1 , M1 ), (Λ2 , α2 , M2 ) −→ Hom f∗ (Λ1 , α1 , M1 ), f∗ (Λ2 , α2 , M2 ) M M Hom(M1 |p , M2 |p ) · p 7→ Hom(f∗ M1 |q , f∗ M2 |q ) · q p∈Λ1 ∩Λ2
q∈f (Λ1 )∩f (Λ2 )
= M
M
¡ ¢ Hom ⊕p|f (p)=q M1 |p , ⊕p|f (p)=q M2 |p
q∈f (Λ1 )∩f (Λ2 )
M¡ ¢ Tp · p 7→ ⊕p|f (p)=q Tp · q.
p
q
The pullback is defined in the same spirit: 0
(f, g)∗ : Ob(FKE τ ) − → Ob(FKE τ ) ¡ ¢ M¡ (p) ∗ ¢ Λ, α, M 7→ Λ , f α, (f |Λp )∗ M , p
where the Λ(p) are the connected components of f −1 (Λ), f |∗Λ(p) M is defined sheaf theoretically. Since Λ(p) is a finite cover of Λ, the pullback is exact, and therefore really defines a vector bundle over Λ(p) . f ∗ α is of course the same for all ¡ components ¢ (p) −1 of the pullback (the Λ all have the same slope) and again as f exp(πiα) = exp(πif ∗ α), such that f ∗ α ∈ (− 12 + k + γ, 12 + k + γ].
7.1 The functors corresponding to n : (x, y) 7→ (−y, x)
103
This yields for morphisms 0
(f, γ)∗ : FKE τ −→FKE τ 0
Mor(FKE τ ) −→ Mor(FKE τ ) ³ ´ ³ ´ M (k) (l) Hom (Λ1 , α1 , M1 ), (Λ2 , α2 , M2 ) −→ Hom (Λ1 , f ∗ α1 , f ∗ M1 ), (Λ2 , f ∗ α2 , f ∗ M2 ) M
k,l
Hom(M1 |p , M2 |p ) · p 7→
p∈Λ1 ∩Λ2
M
M
¡ ¢ Hom (f |Λ()k )∗ M1 |q , (f |Λ()k )∗ M2 |q · q 1
k,l q∈Λ(k) ∩Λ(l) 1 2
= M p
M
M
1
¢ ¡ Hom M1 |f (q) , M1 |f (q)
k,l q∈Λ(k) ∩Λ(l) 1 2
Tp · p 7→
M
M
Tf (q) · q.
k,l q∈Λ(k) ∩Λ(l) 1 2
All linear (complex) symplectic morphisms between two of our complexified symplectic tori are given by composition of the following morphisms µ ¶ r 0 pr , corresponding to ,r ∈ N 0 1 µ ¶ 1 s s- fold Dehn-twist, corresponding to ,s ∈ Z 0 1 µ ¶ −1 0 the involution, corresponding to 0 −1 µ ¶ 0 −1 n, interchanging the coordinates, corresponding to . 1 0 Most of them were already investigated. When we defined πr∗ we in fact defined the operation of (πr , 0), without mentioning it. The functor (πr , n)∗ , n ∈ Z differs from (πr , 0)∗ only by the shift [+n]. The same holds true for the Dehn-twist. For both morphisms g, the angle by which the imaginary axis is turned, is 0. In section 4.5 we found that the symplectic counterpart of the algebraic involution is the functor (ι, 0), where we denote by ι the symplectic involution. ι turns all lines, including the imaginary axis, by ±πi, so that here g = ±1. The two graded morphisms (ι, 1), (ι, −1) are inverse to each other. We had to include a corresponding shift by ±1 to get the counterpart of the algebraic involution. What is left to do, is to describe the pullback and pushforward with respect to n.
7.1
The functors corresponding to n : (x, y) 7→ (−y, x)
We calculate in this section the pushforward and pullback of n : (x, y) 7→ (−y, x) and the algebraic counterpart. In section 8.3.2 we compare it to the functor induced by the Poincar´e-bundle between the derived categories. This example was already mentioned by Polishchuk and Zaslow [63]. We can not confirm their result, however, that the algebraic counterpart is the same as the functor induced by the Poincar´ebundle.
104
Functors Induced by Symplectic Morphisms
First we calculate how n turns the imaginary axis: i = (0, 1) 7→ (−1, 0). This corresponds to a turn of π2 , so our parameter determining the grading lies in 1 1 2 + Z and we are going to investigate (n, + 2 )∗ . First we consider an object of the form ¡ ¢ (−a, t) 12 + k, (V, M ) . For the underlying Lagrangian we get: (−a, t) 7→ (−t, −a) ∼ (−a + t, a), for the grading we get 1 2
+ k 7→ 1 + k = 0 + (k + 1),
so the degree of the object changes by +1. For the local system we note that the monodromy for the system on (−a, t) is with respect to the orientation induced by the parametrization and the orientation of R. It is mapped by n to the local system given by (V, M ) on (−t, −a), still given with respect to the orientation of R. For our objects in FK(E τ ) we choose always the orientation pointing in the direction of the x-axis (see the definition of the Fukaya category in 2) and so we get the local system (V, M −1 ) on the Lagrangian (−a, t) and the induced orientation of R. Now consider an irreducible object L with an underlying Lagrangian Λ with integer slope m > 0 : ¡ ¢ L = (a + t, a(m − 1) + mt), α, (V, M ) . For the underlying Lagrangian we get a (a + t, a(m − 1) + mt) 7→(−a(m − 1) − mt, a + t) ∼ (t, m −
1 m t)
a a a = pm (t, m − t) ∼ pm (− m + t, − m (−2) − t).
For the grading we see that since α ∈ (0 + k, 12 + k) 7→ α ∈ ( 12 + k, 1 + k) = (− 12 + k + 1, k + 1), the grading changes by [+1]. For the local system we note, that since the image of the underlying Lagrangian points in the direction opposite to our usual orientation but is otherwise 1 − 1, we get as local system on the image the bundle given by (V, M −1 ). So far we have calculated: ¡ ¢ ¡ ¢ a a (a + t, a(m − 1) + mt), α, (V, M ) 7→ pm (− m + t, − m (−2) − t), α + 21 , (V, M −1 ) , and we claim:
7.1 The functors corresponding to n : (x, y) 7→ (−y, x)
105
Lemma 7.7. For m > 0 the following equality holds ¡ ¢ ¡ a ¢ a a a pm (− m +t, − m (−2)−t), α+ 12 , (V, M −1 ) = pm∗ (− m +t, − m (−2)−t), (α+ 12 )0 , (V, M −1 ) ,
where (α+ 12 )0 is a suitable real number, lying in the same interval (− 21 +k +1, k +1) as α + 21 . We only have to show, that pm∗ (V, M −1 ) = (V, M −1 ) on the respective Lagrangians. This follows from the fact, that a a a pm : (− m + t, − m (−2) − t) − → (t, m − t)
is a 1-fold covering. For an object for the form ¡ ¢ (a + t, −a), k, (V, M ) with underlying Lagrangian parallel to the x-axis the underlying Lagrangian is mapped as follows (a + t, −a) ∼ (t, −a) 7→ (a, t). The grading does not change, since k ∈ (− 12 + k, 12 + k] 7→ k +
1 2
∈ (− 12 + k, 21 + k].
For to determine the image of the local system we note that the underlying Lagrangian is mapped 1 − 1 to its image and the orientation is still according to our convention. Therefore (V, M ) 7→ (V, M ). Now consider an irreducible object with underlying Lagrangian of integer slope m<0: ¡ ¢ (a + t, a(m − 1) + mt), α, (V, M ) . We denote s = −m. Then we see the underlying Lagrangian is mapped like (a + t, a(−s − 1) − st) 7→(−a(−s − 1) + st, a + t) ∼ (a + t, 1s t) = ps ( as + t, t). If we want to write is with m again, this reads: a (a + t, a(m − 1) + mt) 7→ p(−m) (− m , t).
For the grading we see that since α ∈ (− 21 + k, k) 7→ α +
1 2
∈ (k, k + 12 )
stays se same. For the calculation of the image of the local system we note that the underlying
106
Functors Induced by Symplectic Morphisms
Lagrangian is mapped by n bijectively onto its image, and the image still has our preferred orientation. Therefore (V, M ) 7→ (V, M ). We have calculated for m < 0 so far: ¡ ¢ ¡ ¢ a (a + t, a(m − 1) + mt), α, (V, M ) 7→ p(−m) (− m , t), α + 21 , (V, M ) . Analogously to the case m > 0 we claim: Lemma 7.8. For m < 0 the following equality holds: ¡ ¢ ¡ a ¢ a p(−m) (− m , t), α + 12 , (V, M ) = p(−m)∗ (− m , t), (α + 21 )0 , (V, M ) , with (α + 12 )0 a suitable real number in the same interval (k, k + 21 ) as α + 12 . All that is left to show is that the assertion is correct for the local system. And a 1 p(−m)∗ (V, M ) = (V, M ) since the map p−m : (− m , t) − → (a + t, − m t) is a 1-fold covering. Remark 7.9. The calculations in the general case for underlying Lagrangians with rational slope are similar. When we want to determine the image of the underlying Lagrangian we calculate easily: ¡ ¢ ar n pr (a + t, a(n − 1) + mt) = pm (− ar m + t, − m (−r − 1) − rt). This looks as if the formulas for the general case are an easy generalization from the arbitrary case, but in general we have not a direct analogue of lemmata 7.7 and 7.8. The image of a general object will not be the pushforward along pm but only the pushforward along pm0 for a divisor m0 of m, which might very well be 1: The image of the underlying Lagrangian of course only depends on this Lagrangian, which is essentially determined by the quotient m r . But the image of the whole object depends also on the local system, which depends on the integers r and its rank and not on m. We see this when we calculate the covering n(γpr (Λ) ) − → γn(pr (Λ)) , which is not necessarily one to one but depends on m and r and not only on their quotient. And then we have to find a local system (V, N ), such that pm0 ∗ (V, N ) = (V, M ) which only exists, if the restriction of the map to the components is bijective. We summarize the calculations we have carried out so far in a proposition. Proposition 7.10. Denote by n the map n : E τ −→ E τ , (x, y) 7→ (−y, x). The pushforward of (n, 21 ) is given on irreducible objects with underlying Lagrangians with integer slope m by: ¡ ¢ (n, 12 )∗ (a + t, a(m − 1) + mt), α, (V, M ) ¡ a ¡ ¢´ a 1 0 −1 p (− + t, − (−2) − t), (α + ) , V, M if m > 0, m m 2 ¡ m∗ ¢ = if m = 0, (−a, t), 21 , (V, M ) ´ ¡ a ¡ p 1 0 −1 (− + t, t), (α + ) , V, M if m < 0, (−m)∗
m
2
7.1 The functors corresponding to n : (x, y) 7→ (−y, x)
107
with α + 12 and (α + 12 )0 in the same interval (− 12 + r, 12 + r). Note that if α ∈ (− 12 + k, 0 + k] ⇔ m > 0, the degree d of the object changes from d = k to d = k + 1. The pushforward is for objects with underlying Lagrangian parallel to the y-axis given by ¢ ¡ ¢ ¡ (−a, t), 12 + k, (V, M ) 7→ (−a + t, a), 0 + (k + 1), (V, M ) Before we investigate how this functor maps morphisms, we first mention how the algebraic counterpart of (n, 12 )∗ maps objects. Proposition 7.11. The algebraic counterpart Φ(n, 12 )∗ of the functor (n, 12 )∗ : FK(E τ ) −→ FK(E τ ) maps objects of Db (Eτ ) as follows: An irreducible torsion sheaf S(x, V, N ) is mapped to ¡ ¢ Φ(n, 12 )∗ S(x, V, N ) = L(t∗x ϕ0 ϕ−1 0 ) ⊗ F (V, exp N ). For a special vectorbundle we get: ¡ ¢ Φ(n, 12 )∗ L(t∗−x ϕ0 ϕm−1 ) ⊗ F (V, exp N ) = 0 ¡ ∗ ¢ −2 πm∗ L(tx ψ0 ψ0 ) ⊗ F (V, exp(−N )) [+1] = S(−x, V, N ) ¡ ¢ π(−m)∗ L(t∗x ζ0 ) ⊗ F (V, exp N )
if m > 0 if m = 0 if m < 0.
Here we used the notation ψ0 (z) = ϕ0 (mτ, z) and ζ0 (z) = ϕ0 (−mτ, z). Proof. This is simply an application of the definitions. To calculate the map on the morphism, we first have to explicitly describe the involved spaces: Let ³ ¡ ¢´ Li := (ai + t, ai (mi − 1) + mi t), αi , Mi = Vi , exp(−2πibi + Ni ) . If α2 −α1 6∈ [0, 1] the morphisms space is zero. Therefore consider the case α2 −α1 ∈ (0, 1) we get M Hom(L1 , L2 ) = V2 ⊗ V1∨ ⊗ dk , dk ∈Λ1 ∩Λ2
with dk =
³k + a − a m k + m a − m a ´ 2 1 1 1 2 2 1 , . m2 − m1 m2 − m1
The relation α2 − α1 ∈ (0, 1) implies m2 > m1 . We furthermore suppose 0 < m1 < m2 . Denote by d the greatest common divisor of m1 , m2 , by m0i the quotients m0i = mdi and by m the least common multiple of m1 and m2 . For the target space we get according to proposition 2.20: ˜ 1 ), pm ∗ L ˜ 2) Hom((n, 12 )∗ L1 , (n, 12 )∗ L2 ) = Hom(pm1 ∗ (L 2 gcd(m1 ,m2 )
=
M
M
ν=1
˜ ˜ gs,ν ∈˜ p−1 p−1 1,ν (Λ1 )∩˜ 1,ν (Λ1 )
V2 ⊗ V1∨ · gs,ν ,
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Functors Induced by Symplectic Morphisms
with
¡ ¢´ ¡ a ai −1 1 0 i + t, − (−2) − t), (α + ) , V , M , L˜i = (− m i i mi 2 i
according to proposition 7.10. And with the isomorphism j, explicitly given in lemma 2.19, we calculate ³¡ ¢ 00 ¡ ¢´ 0 0 a1 a1 + t, − (−m − 1) − m t , α −m , V , exp(M ) 1 1 2 2 1 m1 1 ³¡ ¢ ¡ ¢´ (a2 −ν) ∗ 2 −ν) (−m01 − 1) − m01 t , α20 0 , V2 , exp(M2 ) , p˜2,µ L2 = − m2 + t, − (am 2
p˜∗1,µ L1 =
with Mi defined as above, when we calculated the image of an object under n and αi0 0 in the same interval as (αi + 21 )0 . The intersection points are then gs,ν =
³s −
(a2 −ν) a1 m2 + m1 m02 − m01
,
a1 ´ 2 −ν) −m02 s + m02 (am − m01 m 2 1 . m02 − m01
Under the functor (n, 12 )∗ a morphism T · p at an intersection point p is mapped to the morphism T · n(p) at the intersection point n(p). Therefore, we have to determine s, ν, such that n(dk ) = pm (gs,ν ) : ³ −m k − m a + m a k + a − a ´ 2 1 1 1 2 2 1 , , m2 − m1 m2 − m1 ³ m m s − m a + m a + m ν −m s + a − a − ν ´ 2 2 1 1 2 1 2 2 1 1 , , pm (gs,ν ) = m2 − m1 m2 − m1 n(dk ) =
This gives us two equations, which yield k ≡ −m2 s − ν
mod Z(m2 − m1 ),
and there is uniquely one solution ν ∈ Z/dZ and s ∈ Z/(m02 − m01 )Z for k ∈ Z/(m2 − m1 )Z, given by division with rest. Proposition 7.12. The map for (n, 21 )∗ between morphisms spaces in the case 0 < m1 < m2 , α2 − α1 ∈ (0, 1) is given by M ¡ ¢ Hom(L1 , L2 ) = V2 ⊗ V1∨ · dk −→ Hom (n, 12 )∗ (L1 ), (n, 21 )∗ (L2 ) dk ∈Λ1 ∩Λ2
with ¡ ¢ ¡ ¢ ˜ 1 ), pm ∗ (L ˜ 2) Hom (n, 12 )∗ (L1 ), (n, 12 )∗ (L2 ) = Hom pm1 ∗ (L 2 gcd(m1 ,m2 )
=
M
˜ 1 ), p˜∗2,ν (L ˜ 2 )) Hom(˜ p∗1,ν (L
ν=1 gcd(m1 ,m2 )
=
M
M
ν=1
˜ ˜ gs,ν ∈˜ p−1 p−1 1,ν (Λ1 )∩˜ 2,ν (Λ2 )
V2 ⊗ V1∨ · gs,ν ,
7.1 The functors corresponding to n : (x, y) 7→ (−y, x)
109
given by dk 7→ gs,ν , where s, ν are the unique solutions of k = −m2 s − ν, given by division of k by −m2 with rest −ν. If α2 − α1 6∈ (0, 1) the morphism spaces are both zero by definition. The cases remaining to compute are: 1. 0 = m1 < m2 2. m1 < 0 < m2 3. m1 < m2 = 0 4. m1 < m2 < 0 5. Li has underlying Lagrangian parallel to y-axis and Lj has the welldefined slope mj , for i 6= j which we do not work out here. The map is always calculated following the same principle: On the one hand side we have the morphisms space Hom(L1 , L2 ), that is generated by points which we call dk . On the other hand side we are given the space Hom((n, 21 )∗ L1 , (n, 21 )∗ L2 ), where, depending on the slopes of L1 , L2 there may be coverings pm involved, in which case we have to use the base change diagram and the isomorphism j defined in lemma 2.13. This yields the generating points which we call gs,ν . We determine how these points are mapped to each other by solving the equation n(dk ) = pm (g,s,ν ). Although we can compute these generating points to be different from the case cinsidered above, this algorithm yields in the case m1 < 0 < m2 and m1 < m2 < 0 formally the same map. In the remaining cases, the map is even easier. For the description of the pullback, first note that (n, 12 )◦(invt, −1) = (n−1 , − 21 ), where invt denotes the involution (x, y) 7→ (−x, −y). The corresponding functor was described in chapter 4. Therefore (n, − 21 )∗ is the composite of the functors (n, 12 )∗ and (invt, −1)∗ . This proves the following proposition, where we state the most important data again: Proposition 7.13. Denote by n the map n : E τ −→ E τ , (x, y) 7→ (−y, x). The pullback of (n, − 21 ) is given on irreducible objects with underlying Lagrangians with integer slope m by: ¡ ¢ (n, − 12 )∗ (a + t, a(m − 1) + mt), α, (V, M ) ´ ¡ a ¡ a 1 0 p ( + t, (−2) − t), (α − ) , V, M if m > 0, m 2 ¡ m∗ m ¢ = if m = 0, (a, t), − 21 + k, (V, M −1 ) ¡ a ¡ ¢´ p 1 0 −1 ( + t, t), (α − ) , V, M ) if m < 0, (−m)∗
m
2
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Functors Induced by Symplectic Morphisms
with α + 12 and (α + 12 )0 in the same interval (− 12 + r, 12 + r). Note that if α ∈ (− 12 + k, 0 + k] ⇔ m > 0, the degree d of the object changes from d = k to d = k + 1. The pushforward is for objects with underlying Lagrangian parallel to the y-axis given by ¡ ¢ ¡ ¢ (−a, t), 12 + k, (V, M ) 7→ (−a + t, a), 0 + k, (V, M ) The functor (n, − 12 )∗ maps morphisms Hom(L1 , L2 ) =
M
V2 ⊗ V1∨ · dk −→ Hom((n, − 12 )∗ L1 , (n, − 21 )∗ L2 ),
dk ∈Λ1 ∩Λ2
with the latter space given by Hom((n, − 12 )∗ L1 , (n, − 12 )∗ L2 ) = Hom(pm1 ∗ L1 , pm2 ∗ L2 ) gcd(m1 ,m2 )
M
=
Hom(˜ p∗1,µ L1 , p˜∗2,µ L2 )
µ=1 gcd(m1 ,m2 )
=
M
M
µ=1
p−1 ft,µ ∈˜ p−1 2,µ Λ2 1,µ Λ1 ∩˜
V2 ⊗ V ∨ · ft,µ ,
given by dk 7→ ft,µ , where t, µ are the unique solutions of k = m2 t + µ given by division of k by m2 with rest µ. As mentioned before, the algebraic counterpart is not contained in pDb ell. It is a Fourier–Mukai transformation and will be subject of section 8.3.2.
Chapter 8
More Functors Between Derived Categories of Elliptic Curves In the category pDb ell as we have defined it above, we considered various functors that come from relatively easy and natural geometric constructions. These were the pullback and pushforward of morphisms of the underlying complex tori, the dualizing functor, the shift functor, tensoring with a line bundle and translations. But there are more functors between derived categories for which we have an interpretation, and we are still looking for an algebraic counterpart to the functors n∗ and n∗ , with the symplectic map n(x, y) = (−y, x). The following functors are possible extensions of pDb ell by generalizing the morphisms in the category → Db (Eτ ) given by tensoring with an arbitrary 1. FA is the functor Db (Eτ ) − coherent sheaf A on Eτ . This enlarges the group of endomorphisms, which previously only contained equivalences. 2. Fα , where α is a nondegenerate correspondence, roughly speaking this is a one dimensional compact subspace in Eτ × Eτ 0 for a detailed definition see appendix C. We define F to be the functor ¡ ¢ Fα (·) = p1∗ p∗2 (·) . By p1 , p2 we denote the projections from α ⊂ Eτ ×Eτ 0 to the respective factor. (This encases the pushforward and pullback along a morphism f : Eτ − → Eτ0 by taking the correspondence α defined by the graph Γf of the morphism, or its transpose Γ0f .) 3. ΦA : Db (Eτ ) − → Db (Eτ 0 ), where A is a coherent sheaf on Eτ × Eτ 0 and we define ΦA to be the functor L
ΦA (·) = Rpτ 0 ∗ (p∗τ (·) ⊗ A) This extends our item 2. We can recover it, by choosing A to be a torsion sheaf with support contained in C.
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More Algebraic Functors
4. ΦA : Db (Eτ ) − → Db (Eτ 0 ), where A is a element of the derived category of coherent sheaves on Eτ × Eτ 0 and ΦA is the corresponding functor, defined as above. - This is the largest possible extension for which we still know a geometric interpretation of the functor.
8.1 8.1.1
Tensoring with a coherent sheaf Tensoring with a vector bundle
The easiest extension of pDb ell is to tensor with an arbitrary vector bundle instead of just line bundles which we investigated in chapter 4. Denote for a vector bundle A by F = FA the functor FA (F) = F ⊗ A. For F1 , F2 ∈ Db (Eτ ), we have Hom(F1 , F2 ) − → Hom(F1 ⊗ A, F2 ⊗ A) f 7→ f ⊗ 1A For now suppose Fi = L(ϕi ) ⊗ F (Vi , exp Ni ), A = L(ϕ) ⊗ F (V, exp N ), with ϕ = t∗x ϕ0 ϕn−1 , x = aτ + b and ϕi = t∗xi ϕ0 ϕn0 i −1 , xi = ai τ + bi . We want to use the 0 isomorphism ν and express the above morphism as a map between the spaces ¡ ¢ ¡ ¢ ¡ ¢ ∨ ∨ ∨ H 0 L(ϕ2 ϕ−1 − → H 0 L(ϕ2 ϕ−1 1 ) ⊗ V2 ⊗ V1 1 ) ⊗ (V2 ⊗ V1 ⊗ V ⊗ V ), so that we can then apply the mirror functor. Lemma 8.1. The morphism of coherent sheaves given by Hom(F1 , F2 ) −→ Hom(F1 ⊗ A, F2 ⊗ A) s 7→ s ⊗ 1A corresponds to the morphism ¡ ¢ ¡ ¢ ∨ ∨ ∨ H 0 L(ϕ2 ϕ−1 → H 0 L(ϕ2 ϕ−1 1 ) ⊗ (V2 ⊗ V1 ) − 1 ) ⊗ (V2 ⊗ V1 ⊗ V ⊗ V ) s ⊗ T 7→ s ⊗ (T ⊗ ∆), where ∆ is the element of V ⊗ V ∨ that corresponds toPthe identity on V ; with a choice of a basis {ci } of V it is explicitly given by ∆ = i ci ⊗ c∨ i . In other words: the following diagram commutes. Hom(F1 , F2 ) f 7→f ⊗1
²
¡ ¢ Hom F (F1 ), F (F2 )
∼ =
∼ =
¡
¢
/ H 0 L(ϕ ϕ−1 ) ⊗ V2 ⊗ V ∨ 2 1 1
¡
²
fk ⊗T 7→fk ⊗T ⊗∆
¢
/ H 0 L(ϕ ϕ−1 ) ⊗ W, 2 1
where W = V2 ⊗ V1∨ ⊗ V ⊗ V ∨
¡ ¢ ∨ Proof. We start with an element s ⊗ T ∈ H 0 L(ϕ2 ϕ−1 1 ) ⊗ (V2 ⊗ V1 ) and push it around in the following diagram. We use the abbreviations S = N2 ⊗ 1V1∨ − 1V2 ⊗ N1∨ ,
8.1 Tensoring with a coherent sheaf
113
and M = N2 ⊗ 1V1∨ ⊗V ⊗V ∨ − 1V1 ⊗ (−N1∨ ) ⊗ 1V ⊗V ∨ + 1V1 ⊗V2∨ ⊗ N ⊗ 1V ∨ − 1V1 ⊗V2∨ ⊗V ⊗ (−N )∨ . ¢ ¡ ∨ H 0 L(ϕ2 ϕ−1 1 ) ⊗ (V2 ⊗ V1 )
5 s⊗T 7→s⊗T ⊗∆
¡
¢
/ H 0 L(ϕ ϕ−1 ) ⊗ W 2 1
' 1
' 6
² ³ ¡ ¢´ −1 0 H L(ϕ2 ϕ1 ) ⊗ F V2 ⊗ V1∨ , exp(S)
² ¡ ¢ H 0 L(ϕ2 ϕ−1 ) ⊗ F (W, exp M ) 1
' 2
' 4
²
Hom(F1 , F2 )
O
3 s7→s⊗id
/ Hom(F1 ⊗ A, F2 ⊗ A)
We proceed according to the numbering of the arrows: 1. This morphism is given by νϕ ϕ−1 ,N2 −N ∨ defined in Proposition 2.9; in this 2 1 1 case: 2 ⊗1−1⊗N1 ) · T. fk ⊗ T 7→ fk (z − N2πi(n 2 −n1 ) 2. We want to describe the isomorphism ¡ ¢ d : Hom L(ϕ1 ) ⊗ F (V1 , exp N1 ),L(ϕ2 ) ⊗ F (V2 , exp N2 ) ³ ¡ ¢´ ∨ ∼ . = H 0 L(ϕ2 ϕ−1 1 ) ⊗ F V2 ⊗ V1 , exp(N2 ⊗ 1 − 1 ⊗ N1 )
On the fibres, this is the isomorphism Hom(V1 , V2 ) − → V2 ⊗ V1∨ . Choose bases {bi,1 } and {bi,2 } for V1 and V2 . Let M be represented by the matrix (mij ) with respect to these bases. Then Hom(V1 , V2 ) − → V2 ⊗ V1∨ X M = (mij ) 7→ mi,j bi,2 ⊗ b∨ j,1 i,j
¡ ¢ x 7→ w∨ (x)v ←[ v ⊗ w∨ ¡ ¢ A morphism f ∈ Hom L(ϕ1 ) ⊗ F (V1 , exp N1 ), L(ϕ2 ) ⊗ F (V2 , exp N2 ) is a map f : (z, v) 7→ (z, f (z)v), with f (z) ∈ Hom(V1 , V2 ); and such that it satisfies the transformation rule f (z + τ ) = ϕ2 ϕ−1 1 · exp(N2 ) · f (z) exp(−N1 ). ¡ ¢ 2 ⊗1−1⊗N1 Define F to be the image of fk z − N2πi(n under the isomorphism d, 2 −n1 ) then this equation transforms to ¡ ¢ F (z + τ ) = ϕ2 ϕ−1 exp N2 ⊗ exp(−N1 )∨ · F (z), 1 which is the condition for sections in the bundle F2 ⊗ F1∨ .
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More Algebraic Functors
3. Here we map Hom(F1 , F2 ) − → Hom(F1 ⊗ A, F2 ⊗ A) F 7→ F ⊗ id ¡ ¢ ¡ ¢ (z, v) 7→ (z, F (z)v) 7→ (z, v ⊗ w) 7→ (z, F (z) ⊗ w) . 4. So we are looking for the image of F ⊗ id and the analogue of the isomorphism d: ³ ˜ d : Hom L(ϕ1 ) ⊗ F (V1 , exp N1 ) ⊗ L(ϕ) ⊗ F (V, exp N ), ´ L(ϕ2 ) ⊗ F (V2 , exp N2 ) ⊗ L(ϕ) ⊗ F (V, exp N ) ³ ¡ ∨ ∨ − →H 0 L(ϕ2 ϕ−1 1 ) ⊗ F V2 ⊗ V1 ⊗ V ⊗ V , ¢´ exp(N2 ) ⊗ exp(−N1∨ ) ⊗ exp(N ) ⊗ exp(−N ∨ ) . We proceed step by step and start with the space ¡ Hom L(ϕ1 ) ⊗ F (V1 , exp N1 ) ⊗ L(ϕ) ⊗ F (V, exp N ),
¢ L(ϕ2 ) ⊗ F (V2 , exp N2 ) ⊗ L(ϕ) ⊗ F (V, exp N )
Using the property of the tensor product and exp N1 ⊗ exp N = exp(N1 ⊗ 1 + 1 ⊗ N ) this is ³ ¡ ¢ = Hom L(ϕ1 ϕ) ⊗ F V1 ⊗ V, exp(N1 ⊗ 1 + 1 ⊗ N ) , ¡ ¢´ L(ϕ2 ϕ) ⊗ F V2 ⊗ V, exp(N2 ⊗ 1 + 1 ⊗ N ) The analogue of the map as d, only with other fibres yields ³ =H 0 L(ϕ2 ϕϕ−1 ϕ−1 1 )⊗ ¡ ¢ ¡ ¢∨ ´ F (V2 ⊗ V ⊗ V1∨ ⊗ V ∨ , exp N2 ⊗ 1 + 1 ⊗ N ⊗ exp −(N1 ⊗ 1 + 1 ⊗ N ) Using the properties of the tensor product: V ⊗ W ∼ = W ⊗ V and the linearity with respect to scalars we get ³ ¡ ∨ ∨ =H 0 L(ϕ2 ϕ−1 1 ) ⊗ F V2 ⊗ V1 ⊗ V ⊗ V , ¢´ exp(N2 ⊗ 1 − 1 ⊗ N1∨ ) ⊗ exp(N ⊗ 1 − 1 ⊗ N ∨ ) ³ ³ ¡ ∨ =H 0 L(ϕ2 ϕ−1 1 ) ⊗ F V2 ⊗ V1 , ¢ ¡ ¢´ ´ exp(N2 ⊗ 1 − 1 ⊗ N1∨ ) ⊗ F V ⊗ V ∨ , exp(N ⊗ 1 − 1 ⊗ N ∨ )
Locally in one fibre, we have ³ ´ ³ ¡ (z, v ⊗ w) 7→ (z, F (z)v ⊗ w) 7→ z 7→ z, fk (z −
N2 ⊗1−1⊗N1 2πi(n2 −n1 ) )
· T ⊗ ∆w
¢´ ,
P V ∨ where ∆ = dim i=1 ci ⊗ ci for a chosen basis {ci } of V. The corresponding automorphism of V is the identity.
8.1 Tensoring with a coherent sheaf
115
5. According to the assertion, we map ¡ ¢ ∨ ∨ H 0 (L(ϕ2 ϕ−1 → H 0 L(ϕ2 ϕ−1 1 ) ⊗ V2 ⊗ V1 ) − 2 ) ⊗ V2 ⊗ V1 ⊗ V ⊗ V fk ⊗ T 7→ fk ⊗ T ⊗ ∆, In the next point, we finally show that this is the right map. 6. We now apply νϕ
−1 2 ϕ1 ,M
with
M = exp(N2 ) ⊗ exp(−N1∨ ) ⊗ exp N ⊗ exp(−N ∨ ) = exp(N2 ⊗ 1W1 − 1V1 ⊗ N1∨ ⊗ 1V ⊗V ∨ + 1V1 ⊗V2∨ ⊗ N ⊗ 1V ∨ − 1W2 ⊗ N ∨ ),
with W1 = V1∨ ⊗ V ⊗ V ∨ ,and W2 = V1 ⊗ V2∨ ⊗ V. So under ν the section fk ⊗ T ⊗ ∆ is mapped to ¡ ¢ fk z − 2πi(nM2 −n1 ) · (T ⊗ ∆) ¡ exp(N2 ⊗1−1⊗N1∨ )⊗exp(N ⊗1−1⊗N ∨ ) ¢ =fk z − · (T ⊗ ∆). 2πi(n2 −n1 ) This equals ¡ =fk z − because
X
N2 ⊗1−1⊗N1 2πi(n2 −n1 )
¢
· T ⊗ ∆w,
exp(N )ci ⊗ exp(−N )∨ c∨ i =
X
ci ⊗ c∨ i .
(8.1.1)
i
On the symplectic side we first have a look at how the objects are mapped under the symplectic counterpart of the functor FA , with A = L(t∗aτ +b ϕ0 ϕn−1 )⊗ 0 F (V, exp N ). Let F = L(t∗aF τ +bF ϕ0 ϕ0nF −1 ) ⊗ F (VF , exp NF ) ∈ Db (Eτ ) be the algebraic counterpart of an element L of FK(E τ ). Then ΦFA : Φτ F 7→ Φτ F F, and ¡ ¢ Φτ F = (aF + t, aF (nF − 1) + nF t), αF , (VF , exp(−2πibF + NF ) ¡ ¢ Φτ F F = (aF + a + t, (aF + a)(nF + n − 1) + (nF + n)t), α0 , M and ¡ ¢ M = VF ⊗ V, exp(−2πi(bF + b) + NF ⊗ 1V + 1VF ⊗ N ) ¡ ¢ ¡ ¢ = VF , exp(−2πibF + NF ) ⊗ V, exp(−2πib + N ) To determine how ΦFA operates on morphisms, first note that for the two coherent sheaves F1 , F2 , with Fi = L(t∗ai τ +bi ϕ0 ϕn0 i −1 ) ⊗ F (Vi , exp Ni ) the morphism spaces are Hom(Φτ F1 , Φτ F2 ) =
n2 −n X1 −1 k=0
(V2 ⊗ V1∨ ) · ek ,
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More Algebraic Functors
and X1 −1 ¡ ¢ n2 −n Hom Φτ (F1 ⊗ A), Φτ (F2 ⊗ A) = (V2 ⊗ V1∨ ⊗ V ⊗ V ∨ ) · dj , j=0
where the intersection points according to 2.4.1 are: ³k + a − a n k + n a − n − a ´ 2 1 1 1 2 2 1 ek = , n2 − n1 n2 − n1 ³ k + a + a − a − a (n + n)k + (n + n)(a + a) − (n + n)(a + a) ´ 2 1 1 1 2 2 1 dk = , n2 + n − n1 − n n2 + n − n1 − n ³ k + a − a n k + n a − n a + n k + nk + na + n a − na − n a ´ 2 1 1 1 2 2 1 1 2 1 1 2 = , n2 − n1 n2 − n1 ´ ³ k + a − a n k + n a − n a + n k + n(a − a ) 1 1 2 2 1 1 2 1 2 1 ≡ , +a n2 − n1 n2 − n1 ¡ n(a2 −a1 ) ¢ = ek + 0, n2 −n1 + a , This gives us an identification of the intersection points ek and dk , and we obtain that for morphisms ΦF is defined by ¡ ¢ Hom Φτ (F1 ), Φτ (F2 ) Hom(F1 , F2 ) ¡ ¢ ∨ H 0 L(ϕ2 ϕ−1 1 ) ⊗ V2 ⊗ V1
Φτ
/ Ln2 −n1 −1 V ⊗ V ∨ · e 2 k 1 k=0 ΦF
F
² ¡ ¢ H 0 L(ϕ2 ϕ−1 1 ) ⊗W
² / Ln2 −n1 −1
Φτ
k=0
¡ ¢ Hom F (F1 ), F (F2 )
W · dk
¡ ¢ Hom Φτ (F (F1 )), Φτ (F (F2 ))
given by ¡ ¢ C(F1 , F2 )−1 fk ⊗ T Â
/ T · ek _
_
C(F1 , F2
)−1
¡² ¢ fk ⊗ T ⊗∆ Â
¡
² ´ ¢³ ¡ ¢ C(F1 , F2 )−1 fk ⊗ T ⊗ ∆ · dk
/ C F (F1 ), F (F2 )
(8.1.2) with the correction terms
³ ¡ ¡ ¢´ 2¢ ∨ b −b1 a1 −a1 1) ∨ − 1V ⊗ N ∨ 2 C(F1 , F2 ) = exp −πi (an22−a exp N ⊗ 1 − 2πi 1 2 V V ⊗V 1 2 2 −n1 n2 −n1 1 1 n2 −n1
and ³ ¡ ¢ ¡ ¡ 2¢ ∨ a1 −a1 1) ∨ ∨ ∨ C F (F1 ), F (F2 ) = exp −πi (an22−a exp −n1 n2 −n1 N2 ⊗ 1V1 ⊗ 1V ⊗V − 1V2 ⊗ N1 ⊗ 1V ⊗V ´ ¢ 1 + 1V2 ⊗V1∨ ⊗ N ⊗ 1V ∨ − 1V2 ⊗V1∨ ⊗ 1V 1N ∨ − 2πi1V2 ⊗V1∨ nb22 −b −n1
8.1 Tensoring with a coherent sheaf
117
We can still simplify our final expression in 8.1.2: ´ ¡ ¢³ ¡ ¢ C F (F1 ), F (F2 ) C(F1 , F2 )−1 fk ⊗ T ⊗ ∆ ¡ ¢³ ¡ ¡ ¡ ¢¢ 2¢ 1) 1 1 =C F (F1 ), F (F2 ) exp πi (an22−a exp − na22 −a N2 ⊗ 1V1∨ − 1V2 ⊗ N1∨ − 2πi1V2 ⊗V1∨ nb22 −b −n1 −n1 −n1 ´ ¡ ¢ fk ⊗ T ⊗ ∆ ¡ ¢³ ¡ ¡ ¢¢ 2¢ ¡ 1) 1 1 = exp πi (an22−a C F (F ), F (F ) exp − na22 −a N2 ⊗ 1V1∨ − 1V2 ⊗ N1∨ − 2πi1V2 ⊗V1∨ nb22 −b 1 2 −n1 −n1 −n1 ´ ¡ ¢ fk ⊗ T ⊗ ∆ ³ ¡ ¡ ¡ 2¢ (a2 −a1 )2 ¢ a2 −a1 1) exp −πi exp N2 ⊗ 1V1∨ ⊗ 1V ⊗V ∨ − 1V2 ⊗ N1∨ ⊗ 1V ⊗V ∨ = exp πi (an22−a −n1 n2 −n1 n2 −n1 ¢´ 1 + 1V2 ⊗V1∨ ⊗ N ⊗ 1V ∨ − 1V2 ⊗V1∨ ⊗ 1V 1N ∨ − 2πi1V2 ⊗V1∨ ⊗V ⊗V ∨ nb22 −b −n1 ³ ´ ¡ a2 −a1 ¡ ¢¢¡ ¢ ∨ b2 −b1 exp − n2 −n1 N2 ⊗ 1V1∨ − 1V2 ⊗ N1 − 2πi1V2 ⊗V1∨ n2 −n1 fk ⊗ T ⊗ ∆ ¢ ¡ ¢ ¡ a2 −a1 ¡ −b1 1 N2 ⊗ 1V1∨ − 1V2 ⊗ N1∨ − 2πi1V2 ⊗V1∨ nb22 −n ) ⊗ exp na22 −a (N ⊗ 1V ∨ − 1V ⊗ N ∨ ) n2 −n1 −n1 1 ³ ´ ¡ ¡ ¢¢¡ ¢ ∨ b2 −b1 1 ∨ − 1V2 ⊗ N1 − 2πi1V ⊗V ∨ exp − na22 −a N ⊗ 1 f ⊗ T ⊗ ∆ 2 k V 2 −n1 1 n2 −n1 1 ³ ´³ ´ ¡ a2 −a1 ¢ ∨ ¢ = 1V2 ⊗V1∨ ⊗ exp n2 −n1 (N ⊗ 1V ∨ − 1V ⊗ N ) (fk ⊗ T ⊗ ∆ ´ ³ ¢ ¡ a2 −a1 =fk ⊗ T ⊗ exp n2 −n1 (N ⊗ 1V ∨ − 1V ⊗ N ∨ ) ∆ ¡ ¢ 1 2 = exp na22 −a fk ⊗ T ⊗ ∆ −n1
(∗)
= exp
At the star, we used the identity exp(M1 ) ⊗ exp(M2 ) = exp(M1 ⊗ 1 + 1 ⊗ M2 ) with −a1 M1 = − na22 −n (N2 ⊗ 1V1∨ − 1V2 ⊗ N1∨ − 2πi1V2 ⊗V1∨ ) 1 ∨ 1 M2 = − na22 −a −n1 (N ⊗ 1V ∨ − 1V ⊗ N ).
So Φ(⊗A) operates on morphisms as ¡ −a1 ¢2 T · ek 7→ exp na22 −n T ⊗ ∆ · dk . 1 Of course, to describe the functor completely we need to describe it on all objects, including torsion sheaves and arbitrary vector bundles, which we do not work out here. We can already see that if a = 0 then this gives a functor that one could call a generalized Dehn-twist: It is a composition of a¡ Dehn-twist and the ¢ endomorphism of FK(E τ ) that tensors each local system by V, exp(−2πib + N ) and gives each morphism a corresponding extra twist. If a 6= 0 all the discussion about translations applies — we have no good geometric interpretations for the action these translations induce on the underlying Lagrangians. ¡ ¢ Tensoring with an arbitrary vector bundle πr∗ L(ϕ) ⊗ F (V, exp N ) yields more complicated formulas, without changing the general situation.
8.1.2
Tensoring with a torsion sheaf
Let A = S(pA , VA , NA ) be a torsion sheaf on Eτ with dim VA = rA . We define the functor L
FA (·) = · ⊗ A.
118
More Algebraic Functors
Now let F be a vector bundle on Eτ , then L
F ⊗ A = F ⊗ A, which is the torsion sheaf with the fibre Fx ⊗ Ax at x and 0 elsewhere. Since Fx is a free OEτ ,x − module and CJZK/(Z nA ) ⊗CJZK (CJZK)nF ∼ = (CJZK/(Z nA ))nF , we get: FA (F) = S(pA , VF ⊗ VA , 1 ⊗ NA ), with VF the fibre of F. This is of course not irreducible, we get as image as many copies of A as the rank of F. Now consider F = S(pF , VF , NF ) a torsion sheaf. A has the projective resolution d
r L(t∗x ϕ−1 → F (C, 1) − → A, 0 ) −
with x = 12 + 12 τ − pA , d = (t∗x θ(τ, z))rA . We briefly check this assertion: Because d has a rA -fold zero at pA and has no other zeroes, according to the local behavior of holomorphic functions (see for example Forster [20]) locally at a point q 6= pA this sequence looks like: ·1
CJZ − qK − → CJZ − qK − → 0, and at q = pA the sequence looks like CJZ − pA K
·(Z−pA )rA
/ CJZ − qK
/ CJZ − pA K/(Z − pA )rA .
This projective resolution of A yields for the torsion sheaf F = S(pF , VF , NF ) : FA (F) = =
M
d ¡ ¢ r H i F ⊗ L(tx ϕ−1 → F ⊗ L(1) [−i] 0 ) −
M
dpF ¡ ¢ H i S(pF , VF , NF ) − → S(pF , VF , NF ) [−i]
At all fibres at q 6= pF this is an isomorphism and so has the homologies 0. If pF 6= pA also in the fibre at the point pA we have an isomorphism, so in this case F is mapped to the zero sheaf. Now suppose p = pA = pF and denote by R the ring R = CJZ − pK and by I, J the ideals I = (Z − p)rA , J = (Z − p)rF . We know that Ap = R/I, Fp = R/J, because both are defined by ideals, which are generated by one object, because R is a principle ideal domain, (Z − p)rF , resp. rA divides the generating element, which has no further zeroes; it follows the elements (Z − p)rF f resp. RA generate the ideal. Now our sequence reads 0
/ R/J
(Z−p)rA
/ R/J
/0
(8.1.3)
8.2 Functors related to correspondences and we get
119
( R/I if rA < rF R/J if rA ≥ rF ( ¡ ¢ rF −rA R/ (Z − p) if rA < rF 1 H (8.1.3) = R/(J : I) ∼ = R/I if rA ≥ rF .
¡ ¢ H (8.1.3) = R/J /(I/J) ∼ = 0
This yields as image ¡ ¢ FA S(pF , VF , NF ) 0 = S(p, VF , NF ) ⊕ S(p, VF , NF )[−1] S(p, VA , NA ) ⊕ S(p, VS /VA , NF mod NA )[−1]
8.2
if pA 6= pF if p = pF = pA , rA ≥ rF if p = pF = pA , rA < rF
Functors related to correspondences
We distinguish correspondences, which are half dimensional subvarieties of Eτ × Eτ 0 and general correspondences, which are arbitrary dimensional subvarieties of the product. For the standard definitions related to correspondences see appendix C. Lemma 8.2. Let α : Eτ ` Eτ 0 be an irreducible correspondence between the elliptic curves Eτ and Eτ 0 . Then it is the composition of graphs of morphisms of elliptic curves. Proof. If α is nondegenerate, then the projections p1 , p2 onto Eτ and Eτ0 are surjective and proper. Therefore, because α is a one dimensional subvariety, p1 is a finite covering of Eτ . Because C is the universal covering of Eτ , p1 must factor over it and α is isomorphic to an elliptic curve and we have a commutative diagram ∼
/E } } }} p1 }} ² ~}} f
α
Eτ
where f is given by an inclusion of lattices, i.e. is an isogeny maybe composed with a translation. Now with the same argument (and the same elliptic curve E ∼ = α, but maybe given by a different lattice) we get for p2 as well the commutative diagram ∼
/ E0 { { {{ p2 {{ g { ² }{
α
Eτ 0
where g is again the inclusion of lattices maybe composed with a translation. So α is the composition of the graphs of f and g and the isomorphism i between E and E 0 , given either by the identity or a change of lattice: α = [Γg ] ◦ [Γi ] ◦ [Γf ] If α is degenerate and irreducible, then it is isomorphic either to Eτ ×{z 0 } or {z}×Eτ 0 and this is the graph of a map that sends everything to the point z 0 respectively z.
120
More Algebraic Functors
For a nondegenerate correspondence the projections p1 , p2 are proper surjective maps; we can therefore define functors on Db (Eτ ) resp. Eτ 0 corresponding to α by Fα : Db (Eτ ) − → Db (Eτ 0 ) V 7→ pτ 0 ∗ (p∗τ V) and Gα : Db (Eτ 0 ) − → Db (Eτ ) V 7→ pτ ∗ (p∗τ 0 V). The functor Gα = Fα0 with α0 = σ∗ α, and σ is the map that interchanges factors in the product Eτ × Eτ 0 ; therefore we will restrict our considerations to Fα . Since nondegenerate correspondences between elliptic curves are essentially isogenies, these functors yield only the already known direct image and inverse image functors: Suppose α is given by the graph of f : Eτ − → Eτ 0 , an isogeny or a translation. Then for a point z ∈ Eτ 0 we get −1 ∗ pτ 0 ∗ (p∗τ V)|z = H 0 (p−1 τ 0 (z), pτ V|pτ 0 (z)) M ¡ ¢ = H 0 (z, y), p∗τ V|(z,y) y∈f −1 (z)
=
M
Vy
y∈f −1 (z)
This is obviously the direct image under the morphism f . With the same considerations we see that Gα gives us the inverse image. If α is degenerate, then one of the maps pτ and p0τ is not surjective and we cannot define our functors like this. That case is treated in the next section 8.3; it is a special case for the functor corresponding to a sheaf on the cartesian product Eτ × Eτ 0 with support in a one dimensional subspace.
8.3
Functors related to sheaves on the cartesian product Eτ × Eτ 0
Let A be a coherent sheaf on the product Eτ × Eτ 0 , and denote by pτ , pτ 0 the canonical projections. We defined the functors ΦA by the formulas: ¡ ¢ L ΦA : Db (Eτ ) − → Db (Eτ 0 ), ΦA (·) = Rpτ 0 ∗ p∗τ (·) ⊗ A ¡ ¢ L ΨA : Db (Eτ 0 ) − → Db (Eτ ), ΨA (·) = Rpx∗ p∗y (·) ⊗ A . Eτ × Eτ 0 is the complex torus C2 /Λ with the maximal lattice µ ¶ µ ¶ µ ¶ µ ¶ τ 1 0 0 Λ=Z +Z +Z 0 +Z . 0 0 τ 1
8.3 Sheaves on Eτ × Eτ 0
121
For objects in Coh(Eτ ×Eτ 0 ) we can use a similar notation as before. In particular for vector bundles we can use the description by multipliers. We briefly recall the construction: C×C is a cover of Eτ ×Eτ 0 ; we pull back our vector bundle, trivialize over C2 ; the bundle we obtain by factoring the action of the lattice on the trivialized bundle is isomorphic to the bundle we started with; the action of the lattice is given by a multiplier for each of its basis-vectors. Denote by F (V, m(τ,0) , m(1,0) , m0,τ 0 , m0,1 ) the vector bundle with fibre V and the multiplier mγ for γ ∈ {(τ, 0), (1, 0), (0, τ 0 ), (0, 1)}. mγ is an isomorphism of complex vectorspaces mγ : F |(z,z 0 ) = V − → V = F |(z,z 0 )+γ and depends holomorphically on (z, z 0 ) (i.e. depends holomorphically on each of the coordinates z and z 0 on Eτ resp. Eτ 0 ).
8.3.1
Locally free sheaves
Denote by z the coordinate of Eτ and by z 0 the coordinate on Eτ 0 . Let A be an irreducible vector bundle on Eτ × Eτ 0 given by the multipliers m(τ,0),A , m(1,0),A , m(0,τ 0 ),A , and m(0,1),A . They are holomorphically dependent on each of the variables z, z 0 on the fibre VA . Then the restriction of A to the line z 0 = q yields the bundle ˜ A|Eτ ×{q} = F (VA , mτ,A(q) ˜ , m1,A(q) ˜ ) =: A(q), with the multipliers mτ,A(q) ˜ (z) = m(τ,0),A (z, q) m1,A(q) ˜ (z) = m(1,0),A (z, q), which we view as a vector bundle on Eτ via the isomorphism Eτ × {q} ∼ = Eτ . Let F be an element of Coh(Eτ ). Then, because A is locally free, tensoring with A already is exact and: L Rpτ 0 ∗ (p∗τ F ⊗ A) = Rpτ 0 ∗ (p∗τ F ⊗ A). At a fibre at q ∈ Eτ 0 we calculate: ∗ Rk pτ 0 ∗ (p∗τ F ⊗ A)|q = H k (p−1 τ 0 (q), (pτ F ⊗ A)|p−1 (q) ) τ0
(8.3.1)
= H k (Eτ × {q}, p∗τ F|Eτ ×{q} ⊗ A|Eτ ×{q} ) ¡ ¢ ˜ = H k Eτ , F ⊗ A(q) =: W k (q). For now let us assume that F is an indecomposable vector bundle. According to the calculation (8.3.1), the image of F has in degree −k at a point z 0 = q ∈ Eτ the fibre W k (q), so we now have to determine the multipliers of the image bundle. When passing from z 0 = q to q +τ 0 , (resp. to q +1) we get an isomorphism of vector bundles ˜ ˜ + τ 0 ), (resp. A(q) ˜ ˜ + 1)) given by (z, v) 7→ (z, mγ,A (z, q) · v) A(q) − → A(q − → A(q on the fibre VA , γ = (0, τ ) (resp. γ = (0, 1) ); as well we have an isomorphism ˜ ˜ + γ), given by 1 ⊗ mγ,A , which induces by composition an F ⊗ A(q) − → F ⊗ A(q
122
More Algebraic Functors
isomorphism W k (q) − → W k (q + γ). For all W k (q) we fix an isomorphism to a fixed k vectorspace W , given by the ν’s and our standard bases by theta functions. Then the isomorphism given by the multiplier of A becomes an automorphism of W k , holomorphically dependent on z 0 . W k (q)
◦mγ,A
O
/ W k (q + γ) O
∼
Wk
(8.3.2)
∼ mpγ,Φ
A (F )
/ Wk
We call these automorphisms mkτ0 ,ΦA (F) and mk1,ΦA (F) . The image of F und the functor ΦA is the complex which has the following component at position k : ¡ ¢ ΦA (F)p = Fτ W p , mpτ 0 ,ΦA (F) , mp1,ΦA (F) . ˜ Though A is irreducible, A(q) may be reducible (if true for one q, then for all). If ˜ this is the case, then also F ⊗ A(q) decomposes and the components will in general have different degree. Therefore, we might have holomorphic section into this bundle as well as into its dual. This yields a decomposition of the image of F: ΦA (F) ∼ , m0 ) ⊕ F (W 1 , m1 0 , m1 )[−1] = F (W 0 , m0 0 τ ,ΦA (F)
1,ΦA (F)
τ ,ΦA (F)
1,ΦA (F)
To determine the image on torsion sheaves, we can use exact sequences, since the functor ΦA is exact by construction. The functor is defined on morphisms as follows: Let f ∈ Hom(F, G) be a morphism of irreducible elements in the derived category on Eτ . Then f induces a map: ¡ ¢ ¡ ¢ ˜ ˜ H k Eτ , F ⊗ A(q) − → H k Eτ , G ⊗ A(q) (8.3.3) s 7→ (f ⊗ 1) ◦ s. ¡ ¢ This is the stalk ΦA (f ) q , it extends to a morphism of sheaves and is also a map ¡ ¢ ¡ ¢ between the complexes ΦA (F) , ΦA (G) , because the differentials in both complexes are zero. To calculate the induced functor on the symplectic side, we have to find our preferred representatives of the involved morphisms. In this generality we can only say that they are section in this space and therefore given by a linear combination of our basis vectors. Then we can apply the mirror functor to a general linear combination. We can see that a vectorbundle not necessarily induces an equivalence, but if it does the resulting functor ist called a Fourier–Mukai transformation. We have a look at some examples: Example 8.3. Let A = F (C, ϕ0 , 1, t∗y ψ0 , 1) = p∗τ Lτ (ϕ0 ) ⊗ p∗τ 0 L(t∗y ψ0 ), y = aA τ 0 + bA . ˜ = Lτ (ϕ0 ) and for F = L(t∗x ϕ0 ϕn−1 )⊗F (V, exp N ), with n > 0 this yields Then A(q) 0 ˜ F ⊗ A(q) = L(t∗x ϕ0 ϕn0 ) ⊗ F (V, exp N ) ¡ ¢ ˜ H 1 Eτ , F ⊗ A(q) = {0} ¡ ¢ ¡ ¢ ∼ ˜ H 0 Eτ , F ⊗ A(q) = H 0 L(t∗ ϕ0 ϕn ) ⊗ V x
∼ = Cn+1 ⊗ V,
0
8.3 Sheaves on Eτ × Eτ 0
123
the first isomorphism ¡ given by νϕ,N , ¢the second by the choice of our standard basis k fk = t∗ x θ[ n+1 , 0] (n + 1)τ, (n + 1)z 7→ ek ∈ Cn+1 . n+1
Now we have to calculate the multipliers m: e m e (0,τ 0 ),A = t∗y ψ0 ; this gives us a mor∗ ˜ − ˜ phism A(q) → A(q) by (z, v) 7→ (z, ty ψ0 (q)v), and via this the map: ¡ ¢ W 0 = H 0 L(t∗x ϕ0 ϕn0 ) ⊗ V 7→ W 0 (fk ⊗ v) 7→ t∗y ψ0 (q)(fk ⊗ V ) This yields as image of F : Rpτ 0 ∗ (p∗τ F ⊗ A) = L(t∗y ψ0 ) ⊗ Fτ 0 (Cn+1 ⊗ V, 1). It is not hard to see that for A = p∗τ Aτ ⊗ p∗τ 0 Aτ 0 with vector bundles Aτ and Aτ 0 this always happens — the image is a bundle which is a tensor product of Aτ 0 and a trivial bundle of rank deg F · deg Aτ . The mirror functor in this example can be described as follows: Let L = (Λ, α, M) be the symplectic counterpart of F = Lτ (ϕ) ⊗ Fτ (V, exp N ), x = aτ + b. Then Φτ ΦA (L) is the element for which the underlying Lagrangian is ΛA = (aA + t, t) ( if deg F = n ≤ 0 [1, 3] with the grading α ∈ 2 12 1 [− 2 , 2 ] if deg F = n ≥ 0 and M = (Cn+1 ⊗ V, exp(−2πibA ) ⊗ 1). Note that these are all nontransversal. The images of all elements have ΛA as the underlying Lagrangian. Now suppose Fi = L(t∗xi ϕ0 ϕn0 i −1 ) and f ∈ Hom(F1 , F2 ) is one of the basis vectors: £ k ¤¡ ¢ f (z) ⊗ T = fk (z) ⊗ T = t∗x2 −x1 θ n2 −n , 0 (n − n )τ, (n − n )z ⊗ T. 2 1 2 1 1 n2 −n1
Then the expression (8.3.3) specializes to ¡ ¢ ¡ ¢ ˜ ˜ H 0 Eτ , F1 ⊗ A(q) − → H 0 Eτ , F2 ⊗ A(q) ¡ ¢ ¡ ¢ = H 0 L(t∗x1 ϕ0 ϕn0 1 ) ⊗ V1 − → H 0 L(t∗x2 ϕ0 ϕn0 2 ) ⊗ V2 a 7→ f ◦ s given by Eτ − → L(t∗x1 ϕ0 ϕn0 1 ) ⊗ V1 − → L(t∗x2 ϕ0 ϕn0 2 ) ⊗ V2 ¡ ¢ ¡ ¢ z 7→ z, s(z) ⊗ v 7→ z, f (z)s(z) ⊗ T · v. To be able to apply the mirror functor, we therefore need to determine for a basis element £ ¤ s = t∗x1 θ nl1 , 0 (n1 τ, n1 z), n1
124
More Algebraic Functors
th coefficients λi in the following equation: t∗x1 θ n1
£
¤ £ k ¤¡ ¢ (n1 τ, n1 z)t∗x2 −x1 θ n2 −n ,0 (n − n )τ, (n − n )z 2 1 2 1 1
l n1 , 0
n2 −n1
=
nX 2 −1
λi t∗x2 θ
i=0
n2
£
i n2 , 0
¤ (n2 τ, n2 z).
The other example we consider here is of general interest, so that we make it a separate section:
8.3.2
Example: The Poincar´ e bundle
Consider the case τ 0 = τ , denote the projections from Eτ × Eτ to the first and second factor by p1 , p2 , and denote by m : Eτ × Eτ − → Eτ the group law. Define A = m∗ L(ϕ0 ) ⊗ p∗1 L(ϕ0 )∨ ⊗ p∗2 L(ϕ0 )∨ and the corresponding functor denote by ΦA : Db (Eτ ) − → Db (Eτ ). Remark 8.4. A is almost the Poincar´e bundle: The Poincar´e bundle P is a line ˆ where X is an abelian variety, X ˆ its dual variety, X ˆ ∼ bundle on X × X, = Pic0 (X) and is defined up to isomorphism by the properties 1. PX×{α} ∼ =α ˆ 2. P{0}×Xˆ is trivial on X. Denote by d : Eτ − → Eˆτ the isomorphism of Eτ to its dual variety, and by A the bundle defined above. Then (1 × d)∗ P ∼ = A. For more details see for example Mumford [57, chapter II.8] or Griffiths and Harris [34, chapter 2.6]. Denote the coordinates on the two factors by z1 and z2 . Then we can calculate the multipliers for A as A = F (C, exp(−2πiz2 ), 1, exp(−2πiz1 ), 1). ˜ the restriction of A to the line z2 = q, and regard these We denote as before by A(q) bundles as bundles on the first factor Eτ . This yields: ¡ ¢ ˜ = F C, exp(−2πiq) , A(q) ³ ¡ ¢´ ˜ + τ ) = F C, exp −2πi(q + τ ) A(q and ³ ´ ¡ ¢ ¡ ¢ ˜ + 1) = F C, exp −2πi(q + 1) = F C, exp(−2πiq) . A(q ˜ ˜ + τ ) are isomorphic, but the isomorphism will be part of Note that A(q) and A(q the multiplier of the image under the functor Φτ . Polishchuk and Zaslow [63] claimed (without mentioning the degree) that ΦA is the algebraic counterpart of (n, − 21 )∗ . (We calculated the functor (n, 12 )∗ in chapter 7, and (n, − 21 )∗ = (n, 12 )∗ [−1].) To compare these two functors, we start with the
8.3 Sheaves on Eτ × Eτ 0
125
easiest example and first calculate the image of the line bundle L(ϕ0 ) under the functor ΦA . By definition ¡ ¢ ΦA L(ϕ0 ) = Rp2∗ (p∗1 L(ϕ0 ) ⊗ A), so we get as fibre at the point z2 = q: ¡ ¢ ˜ ΦA L(ϕ0 )|q = H 0 Eτ , L(ϕ0 ) ⊗ A(q) ³ ¡ ´ ¢ = H 0 L exp(−2πiq) ⊗L(ϕ0 ) ¡ ¢ = H 0 L(t∗q ϕ0 ) . So the image is a line bundle. From now on we write z for z1 for short. We calculate the multiplier mτ of the image via the commutative diagram: ¢ ¡ H 0 L(t∗q ϕ0 )
basis {t∗q θ(τ,z)}
/C mτ
· exp(−2πiz)
² ¡ ¢ H 0 L(t∗q+τ ϕ0 )
²
basis {t∗q+τ θ(τ,z)}
/ C.
mτ maps therefore 1 ∈ C to: 1 7→ t∗q θ(τ, z) = θ(τ, z + q)
¡ ¢ A.1.2 7→ exp(−2πix)θ(τ, z + q) = exp(−2πiz) exp πiτ + 2πi(z + q)θ(τ, z + q + τ ) ∗ = exp(πiτ + 2πiq)θ(τ, z + q + τ ) = ϕ−1 0 (q)tq+τ θ(τ, z)
7→ ϕ−1 0 (q). The same calculation for q 7→ q + 1 gives us m1 = 1. This yields ¡ ¢ ∨ ∼ ΦA L(ϕ0 ) = L(ϕ−1 0 ) = L(ϕ0 ) . A comparison with proposition 7.11 shows, that in this case the functors agree: Φ(n, − 12 )∗ (L(ϕ0 )) = ΦA (L(ϕ0 )). However, we will see that this is no longer true for more complicated objects: For a special vector bundle F = L(ϕ) ⊗ F (V, exp N ) of positive degree, we get by the same considerations for the fibre of the image at z2 = q: ¡ ¢ ¡ ¡ ¢¢ ΦA L(t∗x ϕ0 ϕn−1 ) ⊗ F (V, exp N ) |q = H 0 Eτ , L(ϕ) ⊗ F (V, exp N ) ⊗ L exp(−2πiq) 0 ¡ ¢ = H 0 Eτ , L(t∗x+q ϕ0 ϕ0n−1 ) ⊗ F (V, exp N ) ∼ = Cn ⊗ V.
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More Algebraic Functors
And we calculate the multiplier mτ for the image bundle via the following commutative diagram. ¡ ¢ H 0 L(t∗x+q ϕ0 ϕn−1 ) ⊗ F (V, exp N ) 0
· exp(−2πiz)
¡ ¢ / H 0 L(t∗x+q+τ ϕ0 ϕn−1 ) ⊗ F (V, exp N ) 0 νt∗
ν t∗ q ϕ,N
q+τ
² ¡ H 0 L(t∗x+q ϕ0 ϕ0n−1 ) ⊗ V
² ¡ H 0 L(t∗x+q+τ ϕ0 ϕn−1 )⊗V 0
k basis {t∗x+q θ[ n ,0](nτ,nz)}
² Cn ⊗ V
ϕ,N
k basis {t∗x+q+τ θ[ n ](nτ,nz)} n
n
mτ
² / Cn ⊗ V
For the standard basis {ek } of Cn this gives us: ek ⊗ v 7→ t∗x+q θ[ nk , 0](nτ, nz) ⊗ v = θ[ nk , 0](nτ, nz + x + q) ⊗ v 7→
n k θ[ n , 0](nτ, nz
+x+q+
N 2πi )
·v
N + x + q + 2πi )·v ¡ ¢ A.9 N = exp πi n1 τ + 2πi n1 (x + q + 2πi ) θ[ k−1 n , 0](nτ, nz + x + q + τ + ¡ 1 ¢ N 7→ exp πi n τ + 2πi n1 (q + x + 2πi ) t∗x+q+τ θ[ k+1 n , 0](nτ, nz) ⊗ v n ¢ ¡ 1 N ) v. 7→ ek−1 ⊗ exp πi n τ + 2πi n1 (x + q + 2πi
7→
exp(−2πiz)θ[ nk , 0](nτ, nz
N 2πi )
·v
This yields ¡ ¢ ¡ 1¢ 1 n n , ΦA L(ϕ) ⊗ F (V, exp N ) = F Cn ⊗ V, M ⊗ (t∗−x ϕ0 ϕ−2 0 ) exp(N ) with the morphism M that sends ek 7→ ek−1 if k > 1 and e1 7→ en . To identify this image with a bundle in preferred notation, we have to determine its determinant bundle and its rank. The latter in obviously n · rank F = n · r and 1¢ 1 ¢rk(M ) ¡ ¡ 1 1 −n n n = det(M )rk(N ) · det (t∗ det M ⊗ (t∗−x ϕ0 ϕ−2 exp(N ) n x ϕ0 ) 0 ) exp(N ) ¡ 1 1 ¢n n n = (−1)r(n−1) det(t∗−x ϕ0 ϕ−2 0 ) exp(N ) rk(N ) = (−1)r(n−1) det(exp(−N )(t∗−x ϕ0 ϕ−2 0 ) r = (−1)r(n−1) (t∗−x ϕ0 ϕ−2 0 ) .
It follows, that ΦA (L(ϕ) ⊗ F (V, exp N )) is a vectorbundle of rank nr and degree r. It therefore follows, that it is of the general form ¡ ¢ ¡ ¡ ¢¢ ΦA L(ϕ) ⊗ F (V, exp N ) ∼ = πn∗ L(t∗y ψ0 ψ0−2 ) ⊗ F V, exp(N ) , and have to determine y such that the determinant bundles agree: ¡ ¢ det πn∗ (t∗y ψ0 ψ0−1 exp N ) = (−1)r(n−1) det(N )(t∗y ψ0 ψ0−2 )rk(N ) = (−1)r(n−1) (t∗y ψ0 ψ0−2 )r !
r = (−1)r(n−1) (t∗−x ϕ0 ϕ−2 0 )
8.3 Sheaves on Eτ × Eτ 0
127
and this yields y = −x +
(n−1) 2 τ
= (− na +
(n−1) 2n )nτ
− b.
Recall that according to proposition 7.11 algebraic counterpart of (n, − 12 )∗ maps such an object as: ¡ ¢¢ ¡ L(ϕ) ⊗ F (V, exp N ) 7→ πn∗ L(t∗−x ψ0 ψ0−2 ) ⊗ F V, exp(−N ) . The fact that the multiplier is exp(−N ) instead of exp N is irrelevant here, since the resulting bundles F (V, exp N ) and F (V, exp(−N )) are isomorphic and we are comparing isomorphism classes only. What is disturbing here is the translation by (n−1) 2 τ.
This results from the general fact, that πn maps our distinguished divisor [ 12 + 12 nτ ] to the divisor [ 12 + 12 τ + n−1 2 τ ], which is not our distinguished divisor on Eτ but its ∗ translate by exactly the amount n−1 2 τ. Because Hom(L(ϕ), tx L(ϕ)) = {0} is x 6∈ Λ, there is not a natural transformation of the functor induced by the Poincar´e bundle to the algebraic counterpart of n∗ . It is surprising, how close these functors are, without being the same. We also see this in the case of negative degree. For a special vector bundle of negative degree F = L(t∗x ϕ0 ϕ0−n−1 ) ⊗ F (V, exp N ) we get for the fibre at z2 = q: ³ ¡ ¢´ ¡ ¢ H 1 F ⊗ L exp(−2πiq) = H 1 L(t∗x+q ϕ0 ϕ0−n−1 ) ⊗ F (V, exp N ) ³¡ ¢∨ ´ ∼ [−1] = H 0 L(t∗x+q ϕ0 ϕ0−n−1 ) ⊗ F (V, exp N ) ¡ ¢ = H 0 L(t∗−x−q ϕ0 ϕn−1 ) ⊗ F (V ∨ , exp −N ∨ ) [−1]. 0 To determine, which bundle this is, we calculate the multipliers as before, according to the following diagram. Denote by y the point y = −x − q for abbreviation ¡ ¢ H 0 L(t∗y ϕ0 ϕn−1 ) ⊗ F (V ∨ , exp −N ∨ ) 0
· exp(−2πiz)
¡ ¢ / H 0 L(t∗y−τ ϕ0 ϕn−1 ) ⊗ F (V ∨ , exp −N ∨ ) 0 νt∗
νt∗ q ϕ,N
q+τ
² ¡ H 0 L(t∗y ϕ0 ϕn−1 )⊗V∨ 0
ϕ,N
² ¡ H 0 L(t∗y−τ ϕ0 ϕn−1 ) ⊗∨ V 0 k basis {t∗y−τ θ[ n ](nτ,nz)} n
k basis {t∗y θ[ n ,0](nτ,nz)} n
² Cn ⊗ V ∨
² / Cn ⊗ V ∨
mτ
The calculation is done like before: ek ⊗ v 7→t∗−x−q θ[ nk , 0](nτ, nz) ⊗ V n £ ¤¡ 7→ t∗−x−q θ nk , 0 nτ, n(z + n
= θ[ nk ](nτ, nz − x − q +
N∨ 2πin )
n∨ 2πi )
¢
·v
·v ∨
7→ exp(2πiz)θ[ nk ](nτ, nz − x − q + N 2πi ) · v ¡ 1 ¢ ∨ A.11 N = exp πi n − 2πi( −x−q + 2πin ) t∗−x−q−τ θ[ k+1 n n , 0](nτ, nz + n ¡ ¢1 1 n exp(N ∨ ) n · v 7→ fk+1 ⊗ t∗x ϕ0 ϕ−2 0 (q)
N∨ 2πi )
·v
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More Algebraic Functors
And so we find ¡ ¢ ¡ ¢ ΦA L(t∗x− n−1 τ ϕ0 ϕ−n−1 ) ⊗ F (V, exp N ) = πn∗ L(t∗y ϕ0 ) ⊗ F (V ∨ , exp −N ∨ ) [−1] 0 2
and we determine y by the same calculation as above to be y =x+
(n+1) 2 τ.
The functor ΦA shares the more astonishing properties of the the algbraic correspondant of n∗ : A bundle of degree zero is mapped to a torsion sheaf and a torsion sheaf is mapped to a bundle of degree zero. Consider for simplicity a line bundle −1 ∗ ˜ F = L(t∗x ϕ0 ϕ−1 0 ). Then F ⊗ A(q) = L(tx+q ϕ0 ϕ0 ). This bundle has nonzero holomorphic sections only if q = −x. Therefore ΦA (F) = S(−x, C, 0). On the other ˜ hand the product S(y, C, 0) ⊗ A(q) has sections for all q and is therefore a vectorbundle, in this specific case a line bundle and we can compute its multiplier to be exp(−2πiy), which yields ΦA (S(y, C, 0)) = L(t∗y ϕ0 ϕ−1 0 ).
8.3.3
Support in a one dimensional subspace
Lemma 8.5. Let α : Eτ ` Eτ 0 be an irreducible nondegenerate correspondence and denote by i the inclusion α ∈ Eτ × Eτ 0 and by A the coherent sheaf i∗ (Oα ). Then Fα = ΦA . L
Proof. By definition ΦA = Rpτ ∗ (p∗τ (·) ⊗ A), with¡ pτ and ¢ pτ 0 the projections from Eτ × Eτ 0 to the respective factors. And Fα = p2∗ p∗1 (·) with p1 , p2 the projections from α ⊂ Eτ × Eτ 0 onto the first resp. the second factor. It suffices to show that for a vector bundle G on Eτ , for each fibre Fα F|y = ΦA G|y , and for each morphism g : G1 − → G2 the fibres at y fulfil Fα g|y = ΦA g|y , because we know that these are exact functors. First suppose that α is the graph of an isogeny f : Eτ − → Eτ 0 . Then Fα is the direct image functor, so M Fα G|y = Gx x|f (x)=y
and for the other functor we get: L
∗ Rpτ 0 ∗ (p∗τ G ⊗ A)|y = H p (p−1 τ 0 (y), (pτ G ⊗ A)|p−1 (y) ) τ0
p
= H (Eτ ×
{y}, (p∗τ G
= H p (Eτ × {y}, =
M
(x,y)∈Eτ ×{y}
¡ ¢ H p x, (G|x )
x:f (x)=y
=
M
x:f (x)=y
⊗ A)|Eτ ×{y} ) M ∩α(G ⊗ A)x )
G|x .
8.3 Sheaves on Eτ × Eτ 0
129
Now suppose α is the graph of the isogeny f : Eτ 0 − → Eτ . Then Fα is the inverse image, so Fα G|y = Gf (y) and for ΦA we calculate: L
ΦA (G)|y = Rpτ 0 (p∗τ G ⊗ A)|y ∗ = H p (p−1 ) τ 0 (y), (pτ G ⊗ A)|p−1 τ (y)
= H p (Eτ × {y}, (p∗τ G ⊗ A)|Eτ ×{y} ) M ¡ ¢ = H p (x, y), (p∗τ G ⊗ A)|(x,y) (x,y)∈Eτ ×{y}∩α
= H p (f (y), G|f (y) ) = Gf (y) . In both cases the image of morphisms come from the morphisms between the respective stalks and are therefore the same for both functors. Let A be a torsion sheaf on Eτ × Eτ 0 with support in an irreducible one dimensional subspace, which we denote by α. α defines a correspondence which we also denote by α : Eτ ` Eτ 0 , and according to lemma 8.2, it is either the composite of graphs of isogenies or of the form {p} × Eτ 0 or Eτ × {q}. Denote by i the inclusion α ⊂ Eτ × Eτ 0 . First assume, α = Γf , with f : Eτ − → Eτ 0 an isogeny. Then the sheaf A is given by A = p∗τ A˜ ⊗ i∗ (Oα ) = p∗τ A˜ ⊗ i∗ OΓf , with a vector bundle A˜ on Eτ . Let F be a coherent sheaf on Eτ ; then L
ΦA (F) = Rpτ 0 ∗ (p∗τ F ⊗ A) L
˜ ⊗ i∗ Oα ) = Rpτ 0 ∗ (p∗τ (F ⊗ A) And this is, according to the above lemma, the composition of the functor that tensors with A˜ and the direct image functor for the morphism f , which we have both already described, as well as their symplectic counterparts. Now suppose the support of A is the graph of a morphism f : Eτ 0 − → Eτ . Then A = p∗τ 0 A˜ ⊗ i∗ Oα with a vector bundle A˜ = F (VA , mA˜) on Eτ0 and ΦA is the composition of the inverse image functor and tensoring with A˜ on Eτ0 . To show that the functors ΦA and the composition of the inverse image functor with the functor ⊗A˜ agree, it suffices to show it for vector bundles and morphisms between them. For a vector bundle F on Eτ we see that ∗ Rpτ 0 ∗ (p∗τ F ⊗ A)|y = H p (p−1 τ 0 (y), (pτ F ⊗ A)|p−1 (y) ) τ0
= = =
H (Eτ × {y}, (p∗τ F ⊗ p∗τ 0 A˜ ⊗ i∗ Oα )|Eτ ×{y} ), ¡ ¢ ˜ H p (f (y), y), (p∗τ F ⊗ p∗τ 0 A)|(f (y), y) ˜ y = W p (y). Ff (y) ⊗ A| p
130
More Algebraic Functors
the multiplicator for passing from W p (y) − → W p (y + τ 0 ) is simply f ∗ mF ⊗ mA˜. So as image of F this yields: ˜ ΦA (F) = f ∗ F ⊗ A. If α = Γf ◦ Γ0g the composition of a graph and an inverse graph, nevertheless, α is isomorphic to an elliptic curve and the functor induced by A is the composition of pullback and pushforward along g and f , tensoring with A on α and maybe we need also the induced functor by some isomorphism, where we identify α with quotients with respect to different lattices. Also these functors were already described extensively. Therefore, the only new functors via this construction, are those that are given by degenerate correspondences {p} × Eτ 0 and Eτ × {q}. First we consider the case {p} × Eτ 0 . Because {p} × Eτ 0 ∼ = Eτ 0 canonically, a sheaf A on {p} × Eτ 0 is just a sheaf in Eτ 0 . For a coherent sheaf F on Eτ , p∗τ F|{p}×Eτ is the constant bundle with fibre Fp so the functor F{p}×Eτ 0 ,A gives for objects dim Fp
F 7→ F (Fp , 1) ⊗ A =
M
A.
k=1
For morphisms we have: Hom(F1 , F2 ) − → Hom(F (F1,p , 1) ⊗ A, F (F2,p , 1) ⊗ A) f 7→ fp ⊗ id . On the symplectic side this gives us the following functor: Denote by (ΛA , αA , MA ), the symplectic counterpart of A and let (Λ, α, M) be an object of FK(E τ ), with algebraic counterpart F. Then ¡ ¢ (Λ, α, M) 7→ ΛA , αA , MA ⊗ (Fp , 1) , and for Li = (Λi , αi , Mi ), two objects of FK(E τ ), with algebraic counterparts Fi ³¡ ¢ ¡ ¢´ Hom(L1 , L2 ) − → Hom ΛA , αA , MA ⊗ (F1,p , 1) , ΛA , αA , MA ⊗ (F2,p , 1) ¡ ¢ ¡ But this space is not defined yet, because ΛA , αA , MA ⊗(F f1,p , 1) , ΛA , αA , MA ⊗ ¢ (F f2,p , 1) are not transversal objects. We have to use here the article of Kreußler [51] for the right definition: ³¡ ¢ ¡ ¢´ Hom ΛA , αA , MA ⊗ (F f1,p , 1) , ΛA , αA , MA ⊗ (F f2,p , 1) ³ ¡ ¢´ =H 0 ΛA , Hom MA ⊗ (F1,p , 1), MA ⊗ (F2,p , 1) ={f : VA ⊗ F2,p − → VA ⊗ F2,p |mA ⊗ 1 ◦ f = f ◦ mA ⊗ 1} with MA = (VA , mA ). And so we get: Hom(F1 , F2 ) Φτ
²
Hom(L1 , L2 )
f 7→fp ⊗id
/ Hom(F1,p ⊗ A, (F2,p ) ⊗ A) Φτ
² ³¡ ¢ ¡ ¢´ / Hom ΛA , αA , MA ⊗ (F f1,p , 1) , ΛA , αA , MA ⊗ (F f2,p , 1) ,
8.3 Sheaves on Eτ × Eτ 0
131
and the arrow in the arrow in the lowest line is calculated by the commutativity of the diagram So for Fi special vector bundle, this yields: fk ⊗ T Â _
/ fk (p)T ⊗ id _
²
² / 1 ⊗ fk (p)T.
C(F1 , F2 )T · ek Â
Now suppose the support of A is the degenerate correspondence α = Eτ ×{q}. Then → Eτ × Eτ 0 the inclusion and A˜ = Fτ (V, mτ ) is a vector A = i∗ A˜ with i : Eτ × {q} − ∼ bundle on Eτ = Eτ × Eτ × {q}. Then for a vector bundle F on Eτ , we calculate 0 ∗ Rpτ 0 (p∗τ F ⊗ A)|z 0 = H 0 (p−1 τ 0 (z ), (pτ F ⊗ A)|p−1 (z 0 ) ) τ0
p
0
}, (p∗τ F
= H (Eτ × {z ( 0 = ˜ H p (Eτ , F ⊗ A)
⊗ A)Eτ ×{z 0 } )
if z 0 6= q if z 0 = q.
˜ So ΦA (F) is a torsion sheaf with support at z 0 = q and global section H p (Eτ , F ⊗ A). To understand which structure the fibre at z 0 = q has as OEτ 0 ,q - module, first notice that since p∗τ F ⊗ A = p∗τ (F ⊗ A˜ ⊗ i∗ OEτ ), with i : Eτ − → Eτ × Eτ0 , z 7→ (z, q), the functor ΦA is the composition of the functor that tensors with A˜ and the functor Φi∗ OEτ . Now it suffices to describe the module structure of Φi∗ OEτ (G) for a sheaf G on Eτ . Let G be locally described by the OEτ ,x − module Mx . (OEτ ×Eτ 0 )(x,y) ⊗(OEτ ×E 0 )(x,y) i∗ OEτ |(x,y) (p∗τ G ⊗ i∗ OEτ )(x,y) ∼ = (p−1 τ G)(x,y) ⊗p−1 τ (OEτ )(x,y) τ ¡ ¢ 0 ∼ = Mx ⊗CJZ−xK (CJZ − xK ⊗C CJZ − yK) ¡ ¢ ⊗CJZ−xK⊗CJZ 0 −yK CJZ − xK ⊗C (CJZ 0 − yK/(Z 0 − q) · CJZ 0 − yK) ∼ = Mx ⊗C (CJZ − yK/(Z 0 − q) · CJZ − yK).
If G is a vector bundle i.e. Mx ∼ = CJZ − xKr , then this decomposes into a direct sum ⊕rk=1 CJZ − xK ⊗ CJZ 0 − yK/(Y − q) · CJZ 0 − yK; this decomposition extends to a decomposition on homology and the action of (Z 0 −q) on the fibre of the image is given by multiplication by 0. So, for a vector bundle F we achieved ΦA (F) = S(q, H p (Eτ , F ⊗ A), 0) F+deg A˜ ⊕deg S(q, C, 0) if deg F + deg A˜ > 0 k=1 ˜ −(deg F+deg A) ⊕k=1 S(q, C, 0)[1] if deg F + deg A˜ < 0 = ˜ =O S(q, C, 0) if deg F + deg A˜ = 0 and det(F ⊗ A) ˜ 6= O 0 if deg F + deg A˜ = 0 and det(F ⊗ A) If F is a torsion sheaf, then the ¢image is still a torsion sheaf with support at q ¡ ˜ and fibre V = H 0 Eτ , S(x, V, N ) , where S(x, V, N ) is the torsion sheaf F ⊗ A.
132
More Algebraic Functors
With the same considerations as above, we see that (Z 0 − q) operates on V as the multiplication by zero and we finally see: ¡ ¢ V ΦA S(x, VF , NF ) = S(q, V, 0) = ⊕dim k=1 S(q, C, 0). Since ΦA is the composition of the functor that tensors with A˜ and the functor Φi∗ OEτ ×{q} it suffices to see that Φi∗ OEτ ×{q} , maps a morphism f ∈ Hom(F1 , F2 ) on Eτ to the induced morphism on homology H 0 (Eτ , F1 ) − → H 0 (Eτ , F2 ) which can be interpreted as a morphism between the corresponding torsion sheaves with support at q and the respective fibres. To describe the functor on the mirror side, we can assume A˜ to be trivial (because ˜ We see that for we already described the functor that tensors with a nontrivial A). an indecomposable object: ¡ ¢ (Λ, α, M) 7→ (0, t), 12 + k + b, (Cn , 1) , with k ∈ Z such that α and 12 + k lie in the same interval ] − 12 + k, k + 12 ] and n and b are determined as follows: let F be the algebraic counterpart of (Λ, α, M), then 0 if F is free and deg F ≥ 0 dim H (Eτ , F) 0 ∨ n = dim H (Eτ , F ) if F is free and deg F < 0 dim H 0 (Eτ , F) if F is a torsion sheaf ( 0 if H 0 (F) 6= 0 b= 1 if H 1 (F) 6= 0. If F is decomposable, we repeat the construction for all its components. Now let ¡ ¢ g ∈ Hom (Λ1 , α1 , M1 ), (Λ2 , α2 , M2 ) be the symplectic counterpart of f ∈ Hom(F1 , F2 ), then f induces a morphism of complexes H(F1 ) − → H(F2 ) which is the image of g under the mirror of Φi∗ OEτ .
8.3.4
Support in a zero dimensional subspace
Irreducible torsion sheaves on Eτ × Eτ 0 with support at a single point cannot be described by the datum of the point (p, q), a C−¡ vectorspace V and ¢ a nilpotent 0 endomorphism N , which describes the action of (Z − p), (Z − q) on V like in the case of the elliptic curve, because OEτ ×Eτ 0 ,(p,q) is not a principal ideal domain. (Note that the ideal defining (p, q) is not principal.) The space of global sections of A is still a complex vectorspace and therefore isomorphic to Cr for an r ∈ N. In likeness to the torsion¡ sheaves ¢on a one dimensional complex torus, we write these torsion sheaves as S (p, q), M , where M is a OEτ ×Eτ0 ,(p,q) ∼ = CJZ − p, Z 0 − qKmodule, which is as complex vectorspace isomorphic to Cr . Let A be an irreducible (torsion) sheaf with support at the point (p, q). For now suppose we can write A = p∗τ Aτ ⊗ p∗τ 0 Aτ 0 for torsion sheaves Aτ on Eτ and Aτ 0 on Eτ0 . Then the tensor product of the projective resolutions of Aτ and Aτ 0 is a projective resolution of A. Suppose we have the projective resolutions d
P −1 − → P0 − → Aτ
8.3 Sheaves on Eτ × Eτ 0
133
and b
Q−1 − → Q0 − → Aτ 0 on the curves, then a projective resolution for A is given by d−2
d−1
p∗τ P −1 ⊗ p∗τ 0 Q−1 −→ (p∗τ P 0 ⊗ p∗τ 0 Q−1 ) ⊕ (p∗τ P −1 ⊗ p∗τ 0 Q0 ) −→ p∗τ P −1 ⊗ p∗τ 0 Q−1 →A τ0 −
(8.3.4)
with the usual differentials: d−2 = (d ⊗ 1Q−1 , 1P −1 ⊗ b) d−1 = −(1P 0 ⊗ b) + d ⊗ 1Q0 Before we proceed in the general theory, here a very simple example, of what this construction looks like: ¡ ¢ Example 8.6. Let Aτ = S(p, C, 0) and Aτ 0 = S(q, C, 0), then A = S (p, q), C, 0 and we get as resolutions : d
Lτ (t∗xp ϕ−1 → Lτ (1) − → Aτ 0 )− b
Lτ 0 (t∗q ψ0−1 ) − → Fτ 0 (C, 1) − → Aτ 0 with xp = − 12 − 12 τ + p, xq = − 12 − 12 τ 0 + q and d : (z, v) 7→ (z, t∗−xp θ(τ, z)v) b : (z, v) 7→ (z, t∗−xq θ(τ 0 , z 0 )v) Now we can use these resolutions to get the following resolution of A. −1 ∗ ∗ p∗τ Lτ (t∗xp ϕ−1 0 ) ⊗ pτ 0 Lτ 0 (txq ψ0 ) d−2
¡ ∗ ¢ ² ¡ ¢ ∗ pτ Lτ (1) ⊗ p∗τ 0 Lτ 0 (t∗xq ψ0−1 ) ⊕ p∗τ Lτ (t∗xp ϕ−1 0 ) ⊗ pτ 0 Lτ 0 (1) d−1
²
p∗τ Lτ (1) ⊗ p∗τ 0 Lτ 0 (1) ²
A Under the canonical isomorphisms with the standard basis on C2 this simplifies to −1 ∗ F (C, t∗xp ϕ−1 0 , 1, txq ψ0 , 1)
¶ µ ∗ −1 ¶ µ 1 0 0 txq ψ0 , 1) − →F (C , −1 , 1, ∗ 0 tx p ϕ 0 0 1
d−2
d−1
2
− →F (C, 1, 1, 1, 1) − →A
134
More Algebraic Functors
with the following formulas for the differentials. Here (z, z 0 ) denotes a point in Eτ × Eτ 0 µ ∗ ¶ ¡ ¢ ¡ t−xp θ(τ, z)v ¢ −2 0 0 d : (z, z ), v 7→ (z, z ), ∗ t−xq θ(τ 0 , z 0 )v µ ¶ ¡ ¡ ¢ v ¢ d−1 : (z, z 0 ), 1 7→ (z, z 0 ), −t∗−xq θ(τ 0 , z 0 )v1 + t∗−xp θ(τ, z)v2 v2 Now we want to describe the functor ΦA , given by such a torsion sheaf A with support at a single point. The functor is defined by the formula ΦA : Db (Eτ ) − → Db (Eτ 0 ) L
F 7→ Rpτ 0 ∗ (p∗τ F ⊗ A). • We already constructed the resolution 8.3.4 for A, which we denote by PA • Li (p∗τ F ⊗ A) = H −i (p∗τ F ⊗ PA ).
If F is a vector bundle on Eτ , then also p∗τ F is torsion free and tensoring with p∗τ F is exact. So we have an exact sequence: 1⊗d−2 ¡ ¢ 0− → p∗τ F ⊗ p∗τ P −1 ⊗ p∗τ 0 Q−1 − → p∗τ F ⊗ (p∗τ P 0 ⊗ p∗τ 0 Q−1 ) ⊕ (p∗τ P −1 ⊗ p∗τ 0 Q0 ) 1⊗d−1
− → p∗τ F ⊗ p∗τ P −1 ⊗ p∗τ 0 Q−1 → p∗τ F ⊗ A − →0 τ0 −
This yields as homologies of the complex 1⊗d−3 ¡ ¢ 0− → p∗τ F ⊗ p∗τ P −1 ⊗ p∗τ 0 Q−1 − → p∗τ F ⊗ (p∗τ P 0 ⊗ p∗τ 0 Q−1 ) ⊕ (p∗τ P −1 ⊗ p∗τ 0 Q0 ) 1⊗d−2
− → p∗τ F ⊗ p∗τ P −1 ⊗ p∗τ 0 Q−1 →0 τ0 −
the sheaves ¡ ¢ • L0 (p∗τ F ⊗ A) = H 0 (p∗τ F ⊗ PA ) = p∗τ F ⊗ A = S (x, y), VF ⊗ VA , 1 ⊗ NA • Lj (p∗τ F ⊗ A) = H −j (p∗τ F ⊗ PA ) = 0 for all j 6= 0
L
So if p∗τ F has no torsion, p∗τ F ⊗ A = p∗τ F ⊗ A. Then we get for the fibres of the higher direct images 0 ∗ Rp pτ 0 ∗ (p∗τ F ⊗ A)|z 0 = H p (p−1 τ 0 (z )(, pτ F ⊗ A)|p−1 (z 0 ) ) τ0 ( 0 if z 0 6= q ¡ ¢ = H p Eτ , S(p, VF ⊗ VA , 1 ⊗ NA ) else ( (VF ⊗ VA , 1 ⊗ NA ) if p = 0, z 0 = q = 0 else
8.3 Sheaves on Eτ × Eτ 0
135
So we have for a vector bundle F on Eτ ΦA (F) = Rp∗τ 0 (p∗τ F ⊗ A) = 0 − → Sτ 0 (q, VF ⊗ VA , 1 ⊗ NA ) − → 0.
Again we see that the resulting images of the functor are not transversal. Now suppose we are given an arbitrary irreducible torsion sheaf A with support at (p, q), with a fibre at (p, q), which is a CJZ − p, Z 0 − qK- module. As complex vectorspace this is isomorphic to Cr and this is in a natural way as well a CJZ − pKand a CJZ 0 − qK- module. We denote the fibre by Cr , Mτ,τ 0 , Mτ or Mτ 0 depending on which module structure we want to consider. Because CJZ − pK and CJZ − pK are principal ideal domains, the module structure is uniquely determined by the data of an nilpotent endomorphism Nτ respectively Nτ 0 on Cr , which defines the action of the generator of the generator of the ideal, which defines the torsion of Mτ respectively Mτ 0 . Now suppose F = F (V, mF ) is a vector bundle on Eτ . Then p∗τ F is a vector bundle on the productspace and tensoring with a vector bundle is exact. Therefore L
p∗τ F ⊗ A = p∗τ F ⊗ A and at a point y ∈ Eτ 0 the image has the fibre Rpτ 0 ∗ (p∗τ F ⊗ A)|y = H p (Eτ × {y}, p∗τ F|Z 0 =y ⊗ A|Z 0 =y ) ( ¡ ¢ H p Eτ , F ⊗ S(x, Cr , Nτ ) if y = q = 0 else ( V ⊗ Cr if y = q for p = 0 = . 0 else So we get for a vector bundle F ΦA (F) = S(q, V ⊗ Cr , 1V ⊗ Nτ 0 ) Consider two vectorbundles F1 , F2 and the morphisms between them. Denote by Vi the fibre of Fi . We have to define the map ¡ ¢ Hom(F1 , F2 ) − → Hom S(q, V1 ⊗ Cr , 1 ⊗ Nτ 0 ), S(q, V2 ⊗ Cr , 1 ⊗ Nτ 0 ) . First note that ³ ¡ ¢´ Hom S q, V1 , 1 ⊗ Nτ 0 , S(q, V2 , 1 ⊗ Nτ 0 ) = {f : V1 ⊗ Cr − → V2 ⊗ Cr |f ◦ (1 ⊗ Nτ 0 ) = f ◦ (1 ⊗ Nτ 0 )} and further that for each f ∈ Hom(V1 , V2 ), f ⊗ 1 is contained in this set. Now we see that for an f ∈ Hom(F1 , F f2 ), we get as image ΦA f the morphism f˜⊗ 1. Where f˜x is the morphism given by composition with fx : F1,x − → F2,x on the homology.
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More Algebraic Functors
8.4
A is a nontrivial extension of a torsion sheaf by a vector bundle or a general element of Db (Eτ × Eτ 0 )
Let A be a coherent sheaf on Eτ × Eτ 0 which is given by a short exact sequence 0− →S− →A− →F − → 0,
(8.4.1)
with a torsion sheaf S and a vectorbundle F ∼ = A/S. Lemma 8.7. Let S be a torsion sheaf on X := Eτ × Eτ 0 and given a short exact sequence 0 −→ S −→ A −→ F −→ 0, with a vector bundle F. If the support of S is zerodimensional, then the extension is trivial. Proof. The equivalence classes of such extensions for a group which is equivalent to Ext1 (S, F). Denote by α the support of F ∨ ⊗ S then Ext1 (F, S) ∼ = H 1 (X, F ∨ ⊗ S) ∼ = H 1 (α, F ∨ ⊗ S|α ) =0 because the support of F ∨ ⊗S is equal to the support of S, which is zero dimensional. And if A is the direct sum of S and F, then also the functor ΦA = ΦS ⊕ ΦF . So we are left with extensions of the form (8.4.1) with an irreducible torsion sheaf S with one dimensional support. Since the map that sends a sheaf to its corresponding functor is functorial, we can calculate ΦS with the knowledge of the exact sequence. Elements in Db (Eτ ×Eτ 0 ) are not simply direct sums of shifted elements of Coh(Eτ × Eτ 0 ), the complexes may have nonzero differentials. So let P • be an element in the bounded derived category. Then we get the functor ΦP j for each nonzero entry in P • . As image of F under ΦP • we get the complex ⊕ΦP j (F)[+j] with the differentials given by the natural transformation ΦP j − → ΦP j+1 induced by the morphism of coherent sheaves P j − → P j+1 : L
ΦP j (F) = Rpτ 0 ∗ (p∗τ F ⊗ P j ) L
ΦP j+1 (F) = Rpτ 0 ∗ (p∗τ F ⊗ P j+1 ), L
L
→ p∗τ F ⊗ P j+2 , a morphism between P j to P j+1 induces a morphism of p∗τ F ⊗ P j − which again induces a morphism on the higher direct images which is unique up to quasi isomorphism.
Chapter 9
More Functors Between Fukaya Categories In this chapter we treat two natural extensions of our category pFKell. In chapter 7 we onlyconsidered functors induced by graded morphisms between the underlying manifolds. - This are not yet the most general objects that operate on the Fukaya category. We can naturally define an operation by a triple (f, γ, V), consisting of 0 a morphisms f : E τ − → E τ , an γ ∈ R, fixing the grading for the image and a 0 vectorbundle V on E τ , by which we twist the local system. We will then further extend the notion of a morphism and replace it by that of a Lagrangian correspondence. We consider as morphisms of the underlying manifold only linear symplectic maps. In the same spirit we only consider linear correspondences. This restriction is necessary to define induced functors on our categories. For the underlying Lagrangian we can use the image, resp. direct sum of the inverse images and we get again a special Lagrangian (or a sum of them). In a further step, we then equip the correspondence with a bundle by which we twist our local systems. The hope of such an extension is of course, to find more functors between the Fukaya categories and maybe find a geometric construction that induces the functors that we have achieved as counterparts of algebraic functors. Unfortunately, this construction does not provide such an explanation, since - as in the complex case correspondences are either degenerate or compositions of graphs of morphisms.
9.1
Functors induced by triples (f, γ, V)
The objects operating naturally on branes and therefore on our Fukaya category are 0 triples (f, γ, V), with a morphisms f : E τ − → E τ , which turns the y- or imaginary axis by the angle πig, a parameter, fixing the grading on the objects in the image, 0 γ ∈ g + Z, and a flat complex vectorbundle V on E τ . (For a more general discussion about branes, Fukaya categories and groups operating on them see for example the book of Seidel [67], part II: Fukaya categories.) We define the functors (f, γ, V)∗ and (f, γ, V)∗ by ¡ ¢ (f, γ, V)∗ (Λ, α, M) = (1E τ 0 , 0, V)∗ (f, γ)∗ (Λ, α, M) ,
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More Symplectic Functors
and analogously ¡ ¢ (f, γ, V)∗ (Λ, α, M) = (f, γ)∗ (1E τ 0 , 0, V)∗ (Λ, α, M) . Since we have already investigated the functors induced by graded morphisms, we only have to deal with the automorphisms (1, 0, V) here. We define (1, 0, V)∗ (Λ, α, M) = (1, 0, V)∗ (Λ, α, M) = (Λ, α, V|Λ ⊗ M). Let V be a local system of rank k on E τ . It then is uniquely determined by the action of the fundamental group of E τ on the fibre of V at the base point of π1 (E τ ). Denote the two generators of the fundamental group with basepoint p by γp,x and γp,y . Then V is given by the maps π1 (E τ , p) × Ck − → Ck , γp,x · v 7→ Mp,x v γp,y · v 7→ Mp,y v with Mp,x , Mp,y ∈ Aut(Ck ), such that Mp,x Mp,y = Mp,y Mp,x . This construction is actually independent of the basepoint we chose: Let p0 be another point of E τ and γ a path connecting p and p0 . Then γ is part of a path in π1 (E τ , p) and we can calculate M ∈ Aut(Ck ) which describes the parallel transport of the fibre Vp to Vp0 along γ. Then π1 (E τ , p0 ) is generated by the elements γp0 ,x = γγp,x γ −1 and γp0 ,y = γγp,y γ −1 and the local system V has the monodromies: Mp0 ,x = M Mp,x M −1 = Mp,x , Mp0 ,y = M Mp,y M −1 = Mp,y . Therefore we can drop the base point in all our formulas. But in fact, we choose as a new base point, the point Λ(0) of a parameterized Lagrangian Λ(t), every time we calculate the restriction of V to this Lagrangian. Example 9.1. Let k = 1, Mx = 1, My = exp(2πiu), for u ∈ R. Then we get for a symplectic counterpart (Λ, α, M) of the special line bundle L(ϕ) ⊗ F (V, exp N ) the following image under the induced functor: First we calculate V|Λ ; note that γΛ : [0, 1] − →R t 7→ (a + t, a(n − 1) + nt) is the positive generator of π1 (Λ) ⊂ π1 (E τ ), given by γ = γy + nγx . Therefore V|Λ = (C, exp(2πinu)) and ³ ¡ ¢´ (1, 0, V)∗ (Λ, α, M) = (Λ, α, (C, exp(2πinu) ⊗ M) = Λ, α, V, exp(2πinu − 2πib + N ) ³ ¡ ¢´ = Λ, α, V, exp(−2πi(b − nu) + N )
9.2 Lagrangian correspondences
139
Note: This functor depends on the slope of the underlying Lagrangian, a phenomenon we have not yet encountered. If (Λ, α, M) is the symplectic counterpart of the line bundle L(ϕ) then the algebraic counterpart to this functor maps L(ϕ) = L(t∗x ϕ0 ϕn−1 ) to L(t∗x−nu ϕ0 ϕn0 ). This is a functor, not lying in pDb ell. Since 0 it is an automorphism, though, we know from theorem 3.1 that there is a sheaf A on Eτ × Eτ inducing the functor.
9.2
Lagrangian correspondences
We have already discussed in chapter 7 that on the one hand we have real symplectic maps between the tori and on the other hand complex symplectic maps. Real symplectic maps are those, which respect the (real) symplectic structure on the tori given by the imaginary part of τ , and complex symplectic maps respect the complexified symplectic structure. As long, as we consider Lagrangians on real two dimensional manifolds, we need not care about whether we want to call those submanifolds where the real symplectic or the complexified symplectic structure vanishes Lagrangians, because the form dx ∧ dy itself is zero on all one dimensional submanifolds. But now we are going to discuss Lagrangians in real four dimensional space and here we do have to make a decision. The graph of a real symplectic map is a real-Lagrangian, while the graph of a complexified symplectic map is complexified-Lagrangian. We have seen before in remark 7.1 that real symplectic maps do not always induce functors on the Fukaya categories. The same is true for real symplectic correspondences. Therefore, because we are mainly interested in the induced functors, not in the maps themselves, we will consider here only the complexified symplectic case. So by Lagrangian we denote from now on a submanifold, on which the whole complexified symplectic structure vanishes, and morphisms are those maps that respect the whole complexified symplectic structure. ¡ ¢ ¡ ¢ 2 , ω = τ dx ∧ dy Definition 9.2. Let E1 = R2 /Z2 , ω1 = τ1 dx¡∧ dy , E2 = R2 /Z 2 2 ¢ be two complexified tori and let L ⊂ T4 := R4 /Z4 , −ω1 + ω2 a linear, compact Lagrangian submanifold. Then we call L a special Lagrangian cycle. We denote the free abelian group generated by irreducible special Lagrangian cycles L(T4 , ω2 − ω1 ) and call its elements special Lagrangian correspondences between the tori. This means a special Lagrangian correspondence is a finite formal linear combination X α= ai [Li ] with irreducible special Lagrangian cycles Li . We denote such a correspondence by α : E1 ` E2 . Like in the algebraic ¡case we define¢ the¡transpose of α: ¢ Denote by 0σ the P map σ : 0 0 τ τ τ τ 0 0 0 0 E ×E − → E ×E , (x, y), (x , y ) 7→ (x , y ), (x, y) . We define α = ai [σ∗ Li ]. α0 is again a Lagrangian correspondence. If the correspondence α consists of just one element α = [L] we also denote the correspondence by L. Remark 9.3. σ is not a symplectic map, but antisymplectic according to our sign convention (M1 , ω1 ) × (M2 , ω2 ) = (M1 × M2 , ω2 − ω1 ). This poses no problem here, since also under antisymplectic maps Lagrangians are transformed into Lagrangians.
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More Symplectic Functors
As usual we want symplectic manifolds with the (special) Lagrangian correspondences as morphisms to form a category. Therefore we have to define the composition of correspondences. As in the classical case, the composition is only defined if certain intersections are transversal. Since we have not defined classes of deformations of Lagrangian correspondences, we cannot achieve a definition of a composition for all triples of objects. This is, how the composition is defined in the good cases: Definition 9.4. For a map p which gives a finite covering of a special Lagrangian cycle onto its image p|L : L − → p(L) we define the proper pushforward p∗ L = np [p(L)], where np ∈ N is the number of sheets of the covering. Definition 9.5. Consider L1 : E τ1 ` E τ2 , L2 : E τ2 ` E τ3 , two irreducible special Lagrangian correspondences that intersect transversally in E τ1 × E τ2 × E τ3 . Then we define their composition L : E τ1 ` E τ3 by the special Lagrangian submanifold ¡ ¢ −1 L = p13∗ p−1 12 L1 t p23 L2 , if the intersection is transversal. The projections in the formula are the following E τ1 × E τ2 ×RE τ3 l
p12 lllll
l lll lv ll
E τ1 × E τ2
p23
²
RRR RRpR13 RRR RRR )
E τ2 × E τ3
E τ1 × E τ3 .
A note on the symplectic structure: We usually define for a product of two symplectic manifolds (Mi , ωi ) the symplectic structure (M1 × M2 , ω2 − ω1 ). So we get two different symplectic manifolds ³ ´ ³ ´ E τ1 × (E τ2 × E τ3 ), ω3 − ω2 − ω1 and (E τ1 × E τ2 ) × E τ3 , ω3 − ω2 + ω1 . Here we do not make a choice of the symplectic structure; we are only interested in the intersection of p∗k,l Li as real submanifolds. In general one has to show that L defined as above, is a special symplectic correspondence. In our case this is not necessary, because we will see that this reduces to the composition of graphes of morphisms, for which we know this is true. For two special Lagrangian correspondences α1 , α2 consisting of several components we can use the same formula: The set theoretic intersection is still defined and as well the covering degree and so on. Definition 9.6. Let L1 ⊂ E τ1 and L2 ⊂ E τ2 be special Lagrangians. Then L1 × L2 is a special Lagrangian in E τ1 × E τ2 and so it defines a special Lagrangian correspondence. A special Lagrangian correspondence that decomposes into a direct product of Lagrangians on the tori is called degenerate. We investigate several examples. Example 9.7. Suppose L1 and L2 are degenerate and given by Lagrangians La ⊂ E τ1 , Lb , Lc ⊂ E τ2 , Ld ⊂ E τ3 , such that L1 = La × Lb , L2 = Lc × Ld . We distinguish three cases
9.2 Lagrangian correspondences
141
−1 1. Suppose Lb = Lc . Then Lb ∩ Lc = Lb and p−1 12 L1 ∩ p23 L2 = La × Lb × Ld . The intersection is still defined set theoretically, but it is not transversal. The projection p13 therefore is not finite and we cannot assign a finite degree to the image La × Ld .
2. Suppose Lb 6= Lc , but they have the same slope, so that Lb ∩ Lc = ∅. Then p−1 12 L1 ∩ p23 L2 = ∅ and, naturally p13 (∅) = ∅. The cycle defined by the pushforward as p13∗ (∅) = 0[∅] has degree 0. 3. Suppose Lb and Lc have different slopes. Then they meet at n different points −1 −1 −1 {pk } and p−1 12 L1 ∩ p23 L2 = La × {pk } × Ld and p13∗ (p1 2 L1 ∩ p23 L2 ) = n[La × Ld ]. Example 9.8. Suppose L1 is degenerate and L2 is the graph of a morphisms f : τ1 E τ2 − → E τ3 . Then L1 = La ©¡ × Lb with Lb ⊂ E τ2 Lagrangians in ¢ª La ⊂τ E and −1 τ3 2 the respective tori and ©¡ ¢ª L2 = y, f (y) ⊂ E ס E ¢. This yields p12 L1 ∩ L2 = La × y ∈ Lb , f (y) . So we get p13 (La × Lb × f Lb ) = La × f (Lb ) is defined set theoretically, and the mapping degree of the projection is the mapping degree of f . The corresponding cycle is therefore deg f · [La × f (Lb )]. Example 9.9. Suppose L1 is the graph of a morphism g : E τ1 − → E τ2 and L2 is the graph morphism f ©¡ : E τ2 − →¢ªE τ3 . So we can describe©¡ the Lagrangians ¢ª by ©¡ of a¢ª −1 L1 = x, g(x) and L2 = y, f (y) ©¡ Now p−1 L ∩ p L = x, g(x), f (g(x)) , 12 1 ¢ª 23 2 −1 −1 and this yields p13 (p12 L1 ∩ p23 L2 ) = x, (f ◦ g)(x) the graph of the composition of the morphisms. And we have a welldefined mapping degree, which is in this case 1. So we have established Γf ◦ Γg = Γf ◦g . The following proposition shows that these are essentially all cases there are. But before we can proof this, we need another lemma. Lemma 9.10. Consider the linear map f : R2 /Z2 −→ R2 /Z2 given by A ∈ GL2 (Z). The mapping degree of f is then deg f = det A. Proof. Let
µ ¶ a b A= . c d
A has the inverse −1
A
1 = det A
µ
¶ d −b . −c a
The inverse image of a point (x, y) ∈ T2 under f is therefore the set { det1 A (dx − by + n, −cx + ay + m)|m, n ∈ Z such that
an+bm cn+dm ad−bc , ad−bc
∈ Z}.
To determine the mapping degree, we have to determine, in how many different ways we can choose (n, m) such that the points det1 A (dx − by + n, −cx + ay + m) lie
142
More Symplectic Functors
in the preimage and are different. First we solve the equations an + bm ∈Z ad − bc cn + dm ∈ Z. ad − bc The solutions set is {n = s1 d − s2 b, m = as2 − cs1 , s1 , s2 ∈ Z}. But of course not all of these pairs give different points det1 A (dx − by + n, −cx + ay + m) ∈ T2 . An easy calculation shows that (s1 , s2 ) and (s01 , s02 ) yield the same point, if there are integers t1 , t2 such that s01 = s1 − (t1 b − t2 a) s02 = s2 + (t1 d − t2 c). So the different solutions are elements in ¡ ¢ Z2 /(s1 , s2 ) ∼ s1 + (t2 a − t1 b), s2 + (t1 d − t2 c) = Z2 /I
(9.2.1)
with I the ideal I = Z(a, −c) + Z(−b, d). What is left to do is to count the elements in this finite set. I defines a sublattice in Z2 and the number of elements in 9.2.1 is equal to the area of a fundamental parallelogram of I. A fundamental parallelogram is spanned by the vectors (a, −c) and (−b, d) and the area spanned by them is equal to their determinant: ¯ ¯ ¯ a −b¯ ¯ ¯ ¯−c d ¯ = ad − bc as claimed. 0
Proposition 9.11. Let L : E τ ` E τ be a irreducible symplectic correspondence. Then one of the following cases occurs: 1.
τ τ0
0
∈ Q and there are symplectic morphisms f : E qτ −→ E τ and g : E pτ −→ E τ with qτ = pτ 0 such that L = Γ0f ◦ Γg ,
0
or 2. The correspondence is degenerate with L = L1 × L2 , where L1 ⊂ E τ , L2 ⊂ E τ are Lagrangians in the respective manifolds.
0
Proof. Denote with p1 , p2 the projection from L to E1 , resp. E2 We distinguish three cases with respect to the dimension of the image of p1 . If the image of p1 is zero dimensional: {pk } = Im(p1 ), then, because L is two dimensional, L = {pk } × E2 . But this is not Lagrangian in T4 , so this cannot happen.
9.2 Lagrangian correspondences
143
S Suppose the dimension of the image is one. Then Im(p1 ) = Lk a union of (Lagrangian) lines in E1 , then, again S due to0 the dimension of L, there exist (Lagrangian) 0 lines Lk in E2 such that L = Lk × Lk . L irreducible yields, L = L1 × L2 . Now suppose p1 is surjective. Denote by p˜1 the projection from T4 − → E1 . Note that −1 −1 p˜1 (z) is compact for a point z ∈ E1 . Since L is compact, p˜1 (z) ∩ L is compact, and, because p1 is surjective, this is a transversal intersection, so it is zero dimensional. This yields: {p−1 ˜−1 1 (z) = p 1 (z) ∪ L} =: q < ∞. Since L is linear, the degree of p1 is constant, so p1 : L − → E1 is a finite covering. Because R2 is the universal covering of E1 , p1 factors over it and L is isomorphic to a torus. We have the following commutative diagram (of differentiable manifolds) (with the same argumentation for p2 ): L>
∼
>> >> > p1 >> >Á
Eτ
/ R2 Z2 Ä Ä ÄÄ ÄÄ f Ä ÄÄ
R2 Z2
∼
@@ @@ @ g @@@ Ã
Eτ
0
/L Ä Ä Ä ÄÄp2 Ä ÄÄ Ä
Therefore, L = Γ0f ◦ Γg . Because L is linear, the maps f and g are of the form R2 R2 − → , (x, y) 7→ A1 (x, y) + V = (a1 x + b1 y + v1 , c1 x + d1 y + v2 ) Z2 Z2 R2 R2 g: 2 − → 2 , (x, y) 7→ A2 (x, y) + W = (a2 x + b2 y + w1 , c2 x + d2 y + w2 ), Z Z f:
with det A1 = q and det A2 = p which is the mapping degree of p2 . This yields for L: L = {(a1 x + b1 y + v1 , c1 x + d1 y + v2 , a2 x + b2 y + w1 , c2 x + d2 y + w2 )} 0
For L to be Lagrangian in E τ × E τ , two tangent vectors v = (a1 v1 + b1 v2 , c1 v1 + d1 v2 , a2 v1 +b2 v2 , c2 v1 +d2 v2 ) and w = (a1 w1 +b1 w2 , c1 w1 +d1 w2 , a2 w1 +b2 w2 , c2 w1 + d2 w2 ) have to fulfill ¯ ¯ ¯ ¯ ¯a1 v1 + b1 v2 a1 w1 + b1 w2 ¯ ¯ ¯ 0 ¯a2 v1 + b2 v2 a2 w1 + b2 w2 ¯ ¯ ¯ (−ω1 + ω2 )(v, w) = − τ ¯ + τ ¯c2 v1 + d2 v2 c2 w1 + d2 w2 ¯ c1 v1 + d1 v2 c1 w1 + d1 w2 ¯ !
= 0. This is equivalent to the assertion: ¯ ¯ ¯ ¯ ¯ ¯ ¯a1 v1 + b1 v2 a1 w1 + b1 w2 ¯ 0 ¯a2 v1 + b2 v2 a2 w1 + b2 w2 ¯ ¯ ¯ +τ ¯ −τ ¯ ¯ c2 v1 + d2 v2 c2 w1 + d2 w2 ¯ c1 v1 + d1 v2 c1 w1 + d1 w2 ¯ ¯ ¯ ¯ ¯v1 w1 ¯ ¯v1 w1 ¯ 0 ¯ ¯ + τ det A2 ¯ = −τ det A1 ¯¯ ¯v2 w2 ¯ v2 w2 ¯ ¯ ¯ ¯ ¯ 0 ¯v1 w1 ¯ = (−qτ + pτ ) ¯ v2 w2 ¯ !
=0 ⇔ qτ = pτ 0
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More Symplectic Functors
and it follows that the quotient τ 0 /τ lies in Q. Because qτ = pτ 0 , we can equip E with the symplectic form qτ dx ∧ dy = pτ 0 dx ∧ dy such that f, g are symplectic maps: 0
Eτ
E qτ = EKpτ KKK g s f sss KKK s KKK ss s % yss
Eτ
0
0
Corollary 9.12. The morphism space of two symplectic manifolds E τ and E τ in pFKell+ contains only degenerate correspondences, if τ /τ 0 ∈ / Q.
Excursion: Real algebraic intersection theory and Lagrangian correspondences Define real submanifolds of dimension k as real k-cycles on a 2n-dimensional manifold M , and define abelian formal groups Ck freely generated by these k-cycles. (The empty set is supposed to lie in all those groups as the zero element.) The free abelian group generated by Lagrangian cycles is a subgroup of Cn . Usually one would like to not work with Ck but with equivalence classes of k-cycles with respect to a good equivalence relation. In an algebraic setting over an algebraically closed field, one considers for example birational equivalence or linear equivalence. In this context maybe cycles that are deformed into each other under Hamiltonian deformations might form a good equivalence class. At least for the half dimensional Lagrangian cycles this is expected to be the appropriate definition, since conjecturally each of them contains either exactly one special Lagrangian cycle or none. The next step would be the definition of a suitable intersection product and then to define pushforwards and pullbacks along morphisms. Because all these operations involve cycles of various (co)-dimension, the subset of Lagrangian cycles is of course not closed under such operations. To make all these definitions in the generic real algebraic or in the symplectic context is hard and worth a study on its own. In our case definitions are easier, because we only have ambient varieties that are manifolds, and our submanifolds are always smooth. The problematic cases for us are those, where our cycles do not meet transversally and we have no means to disturb them, so that they do.
9.3
Functors induced by Lagrangian correspondences An approach
Like morphisms, correspondences should induce functors on the Fukaya categories. Of course we want the functor induced by the graph of a morphism to be the same as the functor induced by the morphism itself. So, unfortunately, this extension of our category pFKell does not provide us with all the functors, we still would like to explain. The only new functors are those that correspond to the degenerate correspondences. Unfortunately it will turn out that not only the definition of a composition is hard for these correspondences, but also the definition of the induced functor ist not clear. But first, we have a look at the nondegenerate case.
9.3 Functors induced by Lagrangian correspondences
9.3.1
145
Nondegenerate Lagrangian correspondences
Analogously to the case of morphisms, we have to make a choice concerning the grading of the objects in the image of our induced functor. Definition and Proposition 9.13. Given an irreducible Lagrangian correspon0 dence L : E τ ` E τ and γ ∈ g + Z, where πig is angle the between the y-axis and its image under L. The pair (L, γ) induces the functors 0
F(L,γ) : FK(E τ ) − → FK(E τ )
and
τ0
G(L,γ) : FK(E ) − → FK(E τ ), which we call pushforward and pullback along L, defined as follows. Denote as before 0 by p1 and p2 the projections from E τ ×E τ on the first and second component. Then for objects: ³ ´ ³ ´ 0 ∗ FL Λ, α, M := p2 (p−1 Λ ∩ L), α , p ˜ p ˜ M , 2∗ 1 1 where α0 is chosen such that it lies in the branch (− 12 + k + γ, 12 + k + γ] if α ∈ (− 21 + k, 12 + k], k ∈ Z and p˜i = pi |L . For two objects we define ¡ ¢ FL : Hom(L1 , L2 ) − → Hom FL (L1 ), FL (L2 ) ³ ´ X T 7→ ⊕y|p2 (y)=dk T |p1 (y) · dk dk ∈FL (Λ1 )∩FL (Λ2 )
For G we define G(L,γ) = F(L0 ,γ) and this yields then P F(L,γ) = G(L0 ,γ) . Now we define for a Lagrangian correspondence ai [Li ] where we choose on each Li and γi the functor Fα = ⊕ai F(Li ,γi ) and aF(L,γ) = ⊕ak=1 F(L,γ) . To check that we have chosen the right definitions, we investigate in two examples that they yield the pushforward and pullback of morphisms in the case where L is a graph. We then have shown that in this case, this really defines functors, because pushforward and pullback are welldefined functors. 0
Example 9.14. Suppose L = Γf with a symplectic morphism f : E τ − → E τ . L is then the set of points n¡ ¢o 0 L = (x1 , y1 ), (x2 , y2 ) = f (x1 , y1 ) ⊂ E τ × E τ . ¡ ¢ Let Λ ⊂ E τ be a Lagrangian in E τ , parameterized by x(t), y(t) . Then: n¡ ¢o 0 p−1 (Λ) = x(t), y(t) × Eτ 1 n¡ ¢o p−1 (Λ) ∩ L = (x(t), y(t)), (x , y ) = f (x(t), y(t)) 2 2 1 n ¡ ¢o 0 p2 (p−1 (Λ) ∩ L) = f x(t), y(t) ⊂ Eτ 1 = f (Λ). So on the Lagrangian the pushforward of f and FL agree. Also for the α and α0 the definitions agree. On the local system we defined the pushforward as the sheaf
146
More Symplectic Functors
theoretic direct image, with the morphism restricted to the underlying Lagrangian. Denote by p˜i the restriction of pi to L. For a sheaf M − → Λ, z ∈ f (Λ) = FL (Λ): ¡ ¢ p˜2∗ p˜∗1 M |z = ⊕y∈L|p2 (y)=z p˜∗1 (M)|y = ⊕y∈L|p2 (y)=z Mp1 (y) = ⊕(y,f (y)=z) p˜∗1 (M)|(y,f (y)=z) = ⊕y∈f −1 (z) My = f∗ M|z . On the morphisms it is obvious that both functors are the same - the definitions agree. 0
Example 9.15. Now suppose L = Γ0f with f : E τ − → E τ a symplectic morphism. Then L is given by the set of points: n¡ ¢o 0 L = (x1 , y1 ) = f (x2 , y2 ), (x2 , y2 ), ⊂ Eτ × Eτ . ¡ ¢ Let Λ ⊂ E τ be a Lagrangian in E τ , parameterized by x(t), y(t) , then the construction yields: n¡ ¢o 0 x(t), y(t) p−1 (Λ) = × Eτ 1 n¡ ¢ ¡ ¢ ¡ ¢o (x(t), y(t)), (x (t), y (t)) |f x (t), y (t) = x(t), y(t) p−1 (Λ) ∩ L = 2 2 2 2 1 0
(d) p2 (p−1 ⊂ E τ |f (Λ(d) ) = Λ 1 (Λ) ∩ L) = ⊕d Λ
= f −1 (Λ). So on the Lagrangian the pullback of f and FL agree. Also for the α and α0 the definitions agree. On the local system we defined the pullback as the sheaf theoretic inverse image, with the morphism restricted to the underlying Lagrangian. Denote by p˜i the restriction of pi to L. For a sheaf M − → Λ, z ∈ Λ(d) ⊂ f (Λ) = FL (Λ): ¡ ¢ p˜2∗ p˜∗1 M |z = ⊕y∈L|p2 (y)=z p˜∗1 (M)|y = ⊕y∈L|p2 (y)=z Mp1 (y) = p˜∗1 (M)|(f (z),z) = Mf (z) = f ∗ M|z . Also in this case for morphisms already the definitions agree.
9.3.2
Degenerate Lagrangian correspondences
The above construction for nondegenerate correspondences does not simply carry over to degenerate correspondences. Suppose we are given a Lagrangian correspon0 dence L = La × Lb with La ⊂ E τ and Lb ⊂ E τ . Denote by sa and sb the slopes and by ya , yb the y− axis intercepts of La and Lb . We want to describe the induced functor as far as possible and outline, where a real definition should head to. Let (Λ, α, M) ∈ F K(E τ ) be an object, given by ¡ ¢ Λ = a + t, a(n − 1) + nt α ∈ ( 21 + k, 12 + k], ¡ ¢ M = V, exp(−2πib + N ) .
9.3 Functors induced by Lagrangian correspondences
147
We have not yet defined a formula for the functor, induced by degenerate correspondences. If we nevertheless apply the formulas for the nondegenerate case, then the underlying Lagrangian of FL (L) is given by ( ∅ if Λ ∩ La = ∅ FL (Λ) = ⊕#Λ∩La Lb else if sa = n and a 6= ya . ∅ = Lb else 1st Case: Λ ∩ La = ∅ : So far we have not yet encountered the empty set, which we defined to be Lagrangian. Its algebraic counterpart is the coherent sheaf 0, which has support in an empty set. These are zero objects in our categories, which are both initial and final. If FL (Λ) = ∅, then the formulas yield for the local system ¡ ¢ FL (M)| = {0}, 0 , because we calculate: (˜ p∗1 M)|(x,y) = Mx = 0 if x ∈ / Λ, and since (x, y) ∈ La × Lb , that means x ∈ La , all x lie outside Λ, and we get the local system with fibre {0}. α is not defined for such an object and we formally write α = ∞. 2nd Case: Λ ∩ La 6= ∅, Λ 6= La : Again we calculate the fibre of the pullback along p1 first: ( 0 if x ∈ / Λ, (˜ p∗1 M)|(x,y) = Mx = V if x ∈ Λ ∩ La Therefore we get FL (M) = (V ⊗s , 1), the trivial sheaf with the fibre V ⊗s , where we denote by s the number of intersection points of Λ and La . FL α = αLb + k + γ, for a γ, that has to be specified for the correspondence. 3rd Case: Λ = La : In this case p˜∗1 M is the sheaf on La × Lb , that is trivial when restricted to a line x × Lb and isomorphic to M, if restricted to La × y. Because the support of p˜∗1 M restricted to p−1 2 (y) is not discrete, we take the homologies along these lines, and since we get all trivial bundles on these lines, we get: FL (M) = (V, 1), the trivial bundle with fibre V . And we can define as before FL α = αLb + k + γ, for a γ, that has to be specified for the correspondence. At first glimpse, the situation does not look to bad so far. We found an elementary extension of the definition on the level of objects and one might think, there might be an equally elementary suitable definition for how to map morphisms, such that this becomes a functor for degenerate correspondences. But this look is deceiving. We have a look at an example, and see how this map looks like on the algebraic side:
148
More Symplectic Functors
Example 9.16. Consider the exact sequence d
0− → L(t∗x ϕ0 ) − → L(1) − → S(p, C, 0) − → 0, with x=
1 2
+ 12 τ − p,
p = aτ + b,
(9.3.1)
¡ ¢ d : (x, v) 7→ x, tx∗ θ(τ, z) .
Let L = L1 × L2 with L1 parameterized by L1 (t) = (t, 3t), and L2 not parallel to the y−axis, such that we have a slope defined. Under the mirror functor Φτ (9.3.1) transforms into the sequence 0 ³ (−a +
1 2
+ t, t),
1 4,
² ¡ ¢´ C, exp(−2πi(−b + 21 )) C(x)·(0,0)
²
¡ ¢ (t, 0), 12 , (C, 0) S(x)·(a,0)
² ¡ ¢ (a, t), 0, (C, 0) ²
0 with the correction terms C(x), S(x) ∈ C, yielded by the definition of Φτ on morphisms. By applying FL so far, we have defined it, this gives us: 0 ³ (−a +
1 2
+ t, t),
1 4,
² ¡ ¢´ C, exp(−2πi(−b + 12 )) Â
FL
¡
¢
¡
¢
¡
¢
/ Lb , αb , (C3 , 0)
C(x)·(0,0)
²
¡ ¢ (t, 0), 21 , (C, 0) Â
FL
/ Lb , αb , (C3 , 0)
FL
/ Lb , αb , (C, 0)
S(x)·(a,0)
² ¡ ¢ (a, t), 0, (C, 0) Â ²
0
9.3 Functors induced by Lagrangian correspondences
149
Applying the mirror functor again in the opposite direction this yields on the algebraic side: 0 ²
L(t∗x ϕ0 ) Â
ΦFL
/
L3
¡ ¢ Lb , αb , (C, 0)
ΦFL
/
L3
¡ ¢ Lb , αb , (C, 0)
−1 k=1 Φτ
d
²
L(1) Â ²
S(p, C, 0) Â
ΦFL
−1 k=1 Φτ
¡
¢
/ Φ−1 Lb , αb , (C, 0) τ
²
0. Note that on the right hand side, we have only vectorbundles. We wish to find a definition such that FL is an exact functor. That means, we have to find a definition such that this becomes an exact sequence of vectorbundles. But an exact sequence of vectorbundles splits - and the dimensions here do not match. It follows that it is not possible to extent the definition of FL to morphisms to get a corresponding functor of the derived category. It follows that we cannot extent the definition of the functor for a correspondence so easily to degenerate correspondences. We somehow have to derive the projection functors first, to make this an exact functor, in analogy to the algebraic case. One would have to change the definition also on the level on objects. It is remarkable that with the definition on objects as it is now, so many objects are mapped to zero - too many it seems. In general, the situation for degenerate correspondences resemble the situation of a torsion sheaf on the product of two tori on the algebraic side. And the objects mapped to zero by the degenerate correspondence, remind of torsion sheaves in the algebraic case. But torsion sheaves are not mapped to the zero object by the algebraic functors corresponding to torsion sheaves on the product, but to a complex which only has zero in degree zero. The definition of a functor corresponding to a Lagrangian correspondence is an interesting problem on its own right, but this leads beyond the scope of this work. Remark 9.17. We treated Lagrangian correspondences in an adhoc and intuitive way, just to give an expression, of what happens for such kinds of functors. While finishing this thesis, the author learned about the preprint [76] by Wehrheim and Woodward, where Lagrangian correspondences are treated in a precise setting. See also references therein.
150
More Symplectic Functors
Chapter 10
Synopsis Section 10.1 provides a summary over all functors we have studied and some conclusions. In section 10.2 we outline interesting questions based on and continuing this work.
10.1
Summary
We started out to extend the mirror functor Φτ : Db (Eτ ) − → FK(E τ ) that we know from Kontsevich [49], Polishchuk and Zaslow [63] and Kreußler [51], to a functor from the 2-category of derived categories of coherent sheaves on elliptic curves to the 2-category of Fukaya categories of complexified symplectic tori. We considered the functor Φ : pDb ellall − → pFKellall where the category pDb ellall denotes the category of pairs (τ, Db (E τ )) with all functors as morphisms, and pFKellall analogously the category of pairs (p, FKE τ ) with all functors as morphism. This functor is trivially an equivalence. Therefore we restricted ourselves to functors that arise via pullback and pushforward from morphisms, shifts and in pDb ell also tensoring with line bundles. We saw that the mirror functor does not extent to a functor between these categories as a restriction of the functor Φ. On the algebraic, as well as on the symplectic category, we found functors without counterpart under the mirror functor. We started in chapter 4 with the following natural automorphisms on the algebraic category. Many of them, though not all, have symplectic counterparts: algebraic functor shift [±1] induced by translations tb general translations t∗aτ +b tensoring with the line bundle L0 = L(ϕ0 ) dualizing induced by z 7→ −z
symplectic counterpart shift [±1] local systems ⊗ exp(2πib) no geometric interpretation Dehn-twist no geometric interpretation induced by (x, y) 7→ (−x, −y)
152
Synopsis
We continued our investigations with a study of the functors induced by isogenies and isomorphisms between complex tori. Only few of them have symplectic counterparts: algebraic functor induced by the isogeny πr induced by the isogeny γs induced by the isomorphism τ 7→ τ + 1 induced by the isomorphism τ → 7 − τ1
symplectic counterpart induced by pr no geometric interpretation no geometric interpretation no geometric interpretation
Then we investigated the functors induced by symplectic morphisms. Most of the resulting functors were already investigated as symplectic counterpart of an algebraic functor. The only exception was n∗ . Here is the complete overview. symplectic functor Dehn-twist induced by pr induced by (x, y) 7→ (−x, −y) induced by (x, y) 7→ (−y, x)
algebraic counterpart tensoring with the line bundle L0 = L(ϕ0 ) induced by πr induced by z 7→ −z algebraic counterpart lies not in pDb ell
We then extended our notion of algebraic functor and investigated various functors that come from more general geometric constructions. We did this in two steps. The first extension included tensoring with arbitrary coherent sheaves (not just line bundles). We generalized the notion of morphisms to correspondences, but showed that this yields no new functors in our case. Then we extended the category a second time by the following construction: We choose a sheaf A on the product of two elliptic curves Eτ × Eτ 0 and defined the functor ΦA induced by A by the formula L
ΦA (·) = Rp2∗ (· ⊗ A). And we found: algebraic functor tensoring with F (V, exp N ) tensoring with an arbitrary vectorbundle tensoring with a torsion sheaf nondegenerate correspondences ΦA , A torsion free ΦA , supp A a degenerate correspondence ΦA , supp A a point
symplectic counterpart tensoring local systems by (V, exp N ) no geometric interpretation no geometric interpretation are essentially morphisms; see there generally no geometric interpretation with the exception of n∗ no geometric interpretation no geometric interpretation
We were still missing a lot of symplectic counterparts, so tried to extend also the notion of symplectic functor. We investigated in chapter 9 the functor induced by tensoring with a local system on E τ and the possibilities to define a functor corresponding to a Lagrangian correspondence. In a last step we thought about equipping our Lagrangian correspondence with a local system.
10.1 Summary
symplectic functor (1, 0, V)∗
induced by a nondegenerate correspondence induced by a degenerate correspondence induced by correspondences equipped with local systems
153
comment defines an automorphisms if V has rank one. Then the algebraic counterpart lies in pDb ell+ . For arbitrary rank conjectured to lie in pDb ell+ as well, but we have not explicitly constructed a counterpart. these are essentially morphisms; see discussion there once we found the correct definition, this might correspond to functors in pDb ell+ would be very nice if we could define this. It resembles the algebraic construction ΦA for sheaves on the product.
One result of these computations is that the mirror functor as defined by Polishchuk, Zaslow [63] and Kreußler [51] should rather be called “mirror symmetry for elliptic curves with the choice of an oriented basis for the lattice of its representation as a complex torus”. It turned out that it depends on various choices, on both the algebraic and the symplectic side, which do not fit well with the structure of pDb ell and pFKell. This is especially apparent when we want to investigate the functors by which those choices differ: Given an elliptic curve E, the choice we have to make on the algebraic side consists of the choice of a lattice and its generators, such that C/(Zτ + Z) ∼ = E. We saw that the maps b and m that describe the difference between two such choices induce functors between the corresponding symplectic categories, for which we have no satisfactory geometric explanation. On the symplectic manifold SL2 (Z) acts as well, the action being generated by the Dehn-twist, where the induced functor corresponds under mirror symmetry to the functor that tensors each coherent sheaf on Eτ with the line bundle L(ϕ0 ), and n, where the algebraic counterpart of n∗ lies in pDb ell+ . The algebraic counterpart is a Fourier–Mukai transformation, but contrary to the claim of Polishchuk and Zaslow [63] not given by the Poincar´e bundle. For the definition of the Fukaya category we had to make a choice of an element of the upper complex half plane τ, which defines in this case the complexified symplectic structure of our real torus. The real part of τ, the B–field is only determined up to Z. Naturally we would want to apply (real-)symplectic maps to E τ , so that E τ ∼ = E τ +1 are naturally isomorphic by means of the identity. (If we consider real symplectic maps, all tori E τ that only differ in the real part of τ are isomorphic.) However, real symplectic maps and in particular the identity do not induce functors between the Fukaya categories (theorem 7.1). So on the symplectic manifold, the construction of the Fukaya category depends on τ in a noncanonical way. We have to understand the equivalence of these categories via the algebraic functor that corresponds to the base change τ 7→ τ + 1, but we have already seen that we do not have a satisfactory explanation for this functor on the symplectic side. Remark 10.1. A similar phenomenon for the algebraic case was observed by Seidel
154
Synopsis
in his construction of mirror symmetry for the quartic surface [68] where he constructs a homomorphism which, although it involves only algebraic objects, cannot be explained without mirror symmetry and the corresponding construction on the symplectic side. In our case the situation on the algebraic and the symplectic category is asymmetric, in particular because we have on the algebraic side the theorem 3.1 of Orlov [59], [60], which allows us to explain all autoequivalences of Db E for a given elliptic curve E geometrically by an object A on E × E; this means for us, we can explain all the equivalences Db (Eτ ) − → Db (Eγ·τ ), for γ ∈ SL2 (Z) and all autoequivalences of Db (Eτ ). Some of the functors between algebraic categories that we constructed by the use of mirror symmetry do not look more complicated than the functors we get by mirror symmetry on the symplectic side. A good example is the following: On the algebraic side, the tensoring with a line bundle L(t∗b ϕ0 ϕ−1 0 ), which is simply the translation by b, is a natural functor, which has an obvious geometric meaning. It induces on the symplectic side a functor that twists each local system by the fixed value exp(−2πib) - this functor does not look especially complicated, but it does not come from a global object. The natural functor on the symplectic side which comes immediately into mind twists local systems by an amount that depends on their slope: To give a local system on E τ , we have determine the monodromy maps for both generators γx , γy of the fundamental group. The positive generator γΛ of a Lagrangian with slope nr , is as element in the fundamental group of E τ given by γΛ = rγx + nγy and therefore, the functor that tensors the local system on Λ with the restriction of the given local system on E τ , twists n-times by the monodromy map for γy and r-times by the monodromy map for γx . When we bring this back to the algebraic side, we are given a functor which tensors an object F by a bundle that depends on the quotient of the degree and the rank of F. Until now the two sides appear symmetric. But if the local system on E τ is a line bundle, then the resulting functor is in an equivalence, and then we know that there is a global object on the algebraic side inducing this functor, even if we cannot construct it. On the symplectic side, however, we do not know how to construct such a global object for the counterpart of t∗b and we do not know whether it exists or not. We generally know very little about functors between and symmetries of Fukaya categories. Therefore it is enlightening to construct these functors using the greater knowledge of the algebraic side.
10.2
Outlook
10.2.1
Families of elliptic curves
Homological mirror symmetry for families of Calabi-Yau manifolds (also deformations, i.e. infinitesimal families) are being studied by various people at the moment (Fukaya [22, 25], Kontsevich and Soibelman [50], Seidel [71] are just some of them). Families of elliptic curves are interesting for several reasons. In particular, a homological mirror symmetry for families might yield a connection between homological mirror symmetry and SYZ mirror symmetry, which assumes the mirror manifolds to be toric fibrations. This is in particular the aim of Kontsevich and Soibelman in
10.2 Outlook
155
[50]. Starting from a naive viewpoint, we discuss in appendix D some thoughts about a category that could be associated to a family of elliptic curves. The general idea is to investigate them in a scheme theoretic setting by means of the functor of points. By this construction and localization we can also view deformations of elliptic curves as local families. Over each family we can view vectorbundles (over the total space), which yields in the local picture deformations of vectorbundles. We show in appendix D that we need all ingredients in this construction to be holomorphic, so that we can not treat paths in H. It would be interesting to investigate whether or not there is a mirror symmetry and an symplectic counterpart for such a construction. And if it exists, how it relates to other constructions mentioned above.
10.2.2
Lagrangian correspondences
It would be interesting to study Lagrangian correspondences in more detail. This it is a by now classical construction in algebraic and analytic geometry and it is interesting to see whether it fits into the context of Floer homology and mirror symmetry. The definition of a map on level of Floer chains and on Floer homology is part of ongoing research. Such a construction was successfully used by Peter Albers [1], and in an still unfinished diploma thesis Felix Schm¨ aschke [65] is constructing a homomorphism between Floer homology for Lagrangian correspondences. The importance of Lagrangian correspondences is reflected in the recent paper of Wehrheim and Woodward [76]. It would be interesting to further extend the notion of morphisms. A natural extension of the functor corresponding to a Lagrangian correspondence would be to equip the Lagrangian correspondence L with a local system L. We use a similar definition as in the algebraic case. For an object the image as the object with underlying Lagrangian given by correspondence as before and for the local system we define ¡ the ¢ ∗ the image as p2∗ p1 M ⊗ L , where we only take the restriction of the projections to the Lagrangian, underlying M and its pullback and so forth, as before. In our case there are no interesting Lagrangian correspondences, so this gives us nothing new, but as it is interesting in general, it might be a good first test to check such a construction in this well understood case of the elliptic curve. In the constructions of Albers or Wehrheim and Woodward it is less obvious, how to adapt the formulas, when equipping the correspondence with a local system.
156
Synopsis
Appendix A
Theta-Functions This appendix contains the material we used about theta functions. Our presentation follows Mumford [58].
A.1
The Riemann theta functions
The classical theta function can be defined by: X θ(τ, z) = exp(πin2 τ + 2πinz). n∈Z
It is an entire function with one period being 1 and the pseudo period τ : θ(τ, z + 1) = θ(τ, z)
(A.1.1)
θ(τ, z + τ ) = exp(−πiτ − 2πiz)θ(τ, z).
(A.1.2)
Note that in our notation this pseudo periodic factor is ϕ0 (τ, z). For every holomorphic function f (z) and real numbers a and b, let (Sb f )(z) = f (z + b) (Ta f )(z) = exp(πia2 τ + 2πiaz)f (z + aτ ). The center of the group of transformations generated by S and T is called the Heisenberg group and is a familiar object in Quantum mechanics. When investigating this group, one is interested in functions, invariant under certain subgroups, which then motivates the definition of theta-functions with characteristics, which turn out to form a basis of the space of functions invariant under those subgroups. We are interested in these theta-functions, because they provide us with a basis of the space of global sections of our line bundles. Definition A.1. With Sb and Ta defined as above, define the theta function by θ[a, b](τ, z) = (Sb Ta θ)[τ, z]. Note that θ(τ, z) = θ[0, 0](τ, z). Later we use these transformations to calculate the zeros of a theta-function. We can express the theta function by the following series, which is sometimes also used as a definition:
158
Theta-Functions
Proposition A.2. X
θ[a, b](τ, z) :=
¡ ¢ exp πi(n + a)2 τ + 2πi(n + a)(z + b)
n∈Z
This can be proven by a simple calculation; it can be found in Mumford’s book [58]. )) is given by the n theta functions Proposition A.3. A basis of H 0 (Lτ (t∗x ϕ0 ϕn−1 0 fk (z) = t∗x θ n
£k
¤
n, 0
(nτ, nz),
k ∈ Z/nZ. Proof. First we show that the theta functions fk are global sections of this line bundle. We have to show 1. fk (z + 1) = fk (z) )(z)fk (z). 2. fk (z + τ ) = (t∗x ϕ0 ϕn−1 0 The first point is immediately clear from the definition. For the second we calculate: k fk (z + τ ) = t∗x θ[ , 0](nτ, n(z + τ )) n n = t∗x [(T k θ(nτ, nz + nτ ))] n
n
∗
= t x [(T k exp(−πinτ − 2πinz)θ(nτ, nz))] n
n
∗
= t x [exp(−πinτ − 2πinz − 2πik)(T k θ(nτ, nz))] n
=
n
t∗x ϕ0 ϕn−1 t∗x θ[ nk , 0](nτ, nz). 0 n
Now these fk have the zeros zjk =
1 1 k j + τ − τ −x+ . 2 2 n n
The sets {zjk |j = 1 . . . n} are different for different k, so the functions fk cannot be linear dependent. From the calculation of zeros we see furthermore that the degree of this line bundle is n, and in this case this is the dimension of its space of global sections. Proposition A.4. Products of theta functions can be calculated according to the Riemann formula: ¡ ¢ ¡ ¢ θ[ na1 , 0] n1 τ, z1 · θ[ nb2 , 0] n2 τ, z2 = X ¡ ¢ ¡ ¢ 2 a−n1 b , 0] n1 n2 (n1 + n2 )τ, n2 z1 − n1 z2 θ[ nn1 d+a+b , 0] (n1 + n2 )τ, z1 + z2 . = θ[ n1nn12nd+n 2 (n1 +n2 ) 1 +n2 d∈Z/(n1 +n2 )Z
A.1 The Riemann theta functions
159
Corollary A.5. Define the following abbreviations nji = ni − nj aji = ai − aj xji =
xi −xj nj −ni .
Then θ
£
¤¡ ¢ £ ¤¡ ¢ n12 τ, n12 (z + x12 ) · θ nb23 , 0 n23 τ, n23 (z + x23 ) = X ¤¡ ¢ £ = λd (x1 , x2 , x3 ) · θ n12 nd+a+b , 0 n τ, n (z + x )) 13 13 13 13 a n12 , 0
Z d∈ n Z 13
with λd (x1 , x2 , x3 ) = θ
£ n12 n23 d+n23 a−n12 b n12 n23 n13
¤¡ ¢ , 0 n12 n23 n13 τ, n12 n23 (x12 − x23 )
Proof. Step by step, we rewrite the Riemann formula: First put n1 z1 and n2 z2 instead of z1 and z2 and get: ¢ ¢ ¡ ¡ θ[ na1 , 0] n1 τ, n1 z1 · θ[ nb2 , 0] n2 τ, n2 z2 = X ¢ ¡ 2 a−n1 b θ[ n1nn12nd+n , 0] n n (n + n )τ, n n z − n n z 1 2 1 2 1 2 1 1 2 2 (n +n ) 2 1 2 d∈Z/(n1 +n2 )Z
¢ ¡ θ[ nn1 d+a+b , 0] (n1 + n2 )τ, n1 z1 + n2 z2 1 +n2
Now put z1 = z + x12 , z2 = z + x23 : ¡ ¢ θ[ na1 , 0] n1 τ, n1 (z + x12 ) · θ[ nb2 , 0](n2 τ, n2 (z + x23 )) = X ¡ ¢ 2 a−n1 b θ[ n1nn12nd+n , 0] n1 n2 (n1 + n2 )τ, n1 n2 (x12 − x23 ) 2 (n1 +n2 ) d∈Z/(n1 +n2 )Z
¡ θ[ nn1 d+a+b , 0] (n1 + n2 )τ, (n1 + n2 )(z + +n 1 2
¢
n1 x12 −n2 x23 ) n1 +n2
x −x
Finally put xij = njj −nii and put into the formula n2 − n3 for n1 and n3 − n2 for n2 and get the assertion. Now we state and proof some more formulas we need in this work. The following lemma was pointed out to me by Tobias Finis: Lemma A.6. n−1 X a=0
θ[ na , 0](nτ, nz) = θ( nτ , z)
160
Theta-Functions
Proof. n−1 X
θ[ na , 0](nτ, nz)
=
n−1 X
X
exp(πi(m + na )2 nτ + 2πi(m + na )nz)
a=0 m∈Z
a=0
=
X
2
exp(πi m nτ + 2πi m n nz) n2
m∈Z
= =
X
exp(πim2 nτ + 2πimz)
m∈Z θ( nτ , z)
Lemma A.7. For
µ ¶ a b ∈ SL2 (Z) c d
the theta-function transforms according to the formula 1
2
cz +b z 2 θ( aτ cτ +d , cτ +d ) = ζ(cτ + d) exp(πi cτ +d )θ(τ, z),
(transformation formula)
for an appropriate choice of ζ with ζ 8 = 1. For the proof see the book of Mumford [58]; it is there theorem 7.1. Corollary A.8. Specifically the following identities are true: θ(− nτ , − nz τ +y−
m τ )
1
τ 2 = exp(− πi 4 )( n ) exp(
yτ m πi(z− n + n )2 n )θ( nτ , z τ
−
yτ n
+
m n ),
with the square root chosen in the first quadrant of the complex plane. And: θ(nτ, −nτ z + x −
N 2πi
1
τ 2 + mτ ) = exp(− πi 4 )( n ) exp(−πinτ (−z +
x nτ
1 θ(− nτ , −z +
N m 2 2πinτ + n ) ) x N m nτ − 2πinτ + n ).
−
Proof. Use lemma A.7 with µ ¶ µ ¶ a b 0 −1 = , c d 1 0 m and nτ instead of τ , and for z put in −z + yτ n − n ; then use that θ is an even function. The second formula follows from the first by inserting − τ1 for τ and −z for z.
Lemma A.9. θ[ nk , 0](nτ, nz) = exp(πi n1 τ + 2πiz)θ[ k−1 n , 0](nτ, nz + τ ). Remark A.10. This formula differs from (A.1.2), in that we add there nτ to the argument, here we only add τ.
A.2 The meromorphic functions with two periods
161
Proof. θ
£ k−1
¤ , 0 (nτ, nz + τ ) X ¡ 2 = exp πi(m + k−1 n ) nτ + 2πi(m + n
k−1 n )(nz
¢ + τ)
m∈Z
=
X
exp(πi(m + nk )2 nτ + 2πi(m + nk )(nz + τ ) − 2πi(m + nk )τ + πi n1 τ − 2πi n1 (nz + τ ))
m∈Z
= exp(−πi n1 τ − 2πiz) =
exp(−πi n1 τ
−
X
exp(πi(m + nk )2 nτ + 2πi(m + nk )nz)
m∈Z £ ¤ 2πiz)θ nk , 0 (nτ, nz)
Corollary A.11. θ[ nk , 0](nτ, nz) = exp(πi n1 τ − 2πiz)θ[ k+1 n , 0](nτ, nz − τ ).
A.2
The meromorphic functions with two periods
We briefly report on the meromorphic, doubly periodic functions. The main reference is the book of Hurwitz [40]. We start with some definitions: Let Λ = ω1 Z+ω2 Z be a fixed (maximal) lattice in C. All products and sums without further index run over all ω ∈ Λ; a prime at the sum or product means it runs over all ω ∈ Λ \ 0. Definition and Proposition A.12. Define the following functions Y ¡ ¢ 0 σ(z) = z (1 − ωz ) exp( ωz + 21 ( ωz )2 ) 1 z ¢ 1 X 0¡ 1 + + 2 ζ(z) = + z z−ω ω ω X ¡ ¢ 0 1 1 ℘(z) = z 2 + − ω12 (z−ω)2 ℘ can be expressed by σ via the differential equations ℘(z) = −
d σ 0 (z) . dz σ(z)
Define η1 = 2ζ( ω21 ), η2 = 2ζ( ω22 ) and for ω = m1 ω1 + m2 ω2 ∈ Λ define η(ω) = m1 η1 + m2 η2 . While ℘ is periodic, σ and ζ are quasi-periodic; for ω ∈ Λ: ℘(z + ω) = ℘(z) ζ(z + ω) = ζ(z) + η(ω) σ(z + ω) = (−1)m1 +m2 +m1 m2 exp(η(ω)(z + ω2 ))σ(z). Proposition A.13. Define g2 , g3 by the following series X 1 0 g2 = 60 ω4 X 1 0 g3 = 140 , ω6 then we can write the functional equation for ℘: (℘0 )2 = 4℘3 − g2 ℘ − g3 .
(A.2.1)
162
Theta-Functions
℘ and these coefficients have a multitude of other interpretations. For example they turn up again in the equation for the elliptic curve that can be embedded in projective space via ℘ and therefore are part of the determinant of this curve, which can also be defined as elliptic function and so on. We will only mention, what we really need here. For further readings see [40],[17],[46] or similar. Proposition A.14 ([40]II,1,13 theorem 1). The powerseries for σ at z = 0 is of the form σ(z) = z + k2 u5 + k3 u7 + · · · , where the coefficients are entire rational functions of g2 and g3 with rational coefficients. In particular g2 k2 = − , 240 g3 k3 = − . 840 From this we see that σ is an entire function, with its zeros, all of which are simple, in Λ. We are interested in σ because every elliptic function can be expressed as a fraction of σ-functions. Recall that for an elliptic function the number of zeros equals the number of poles in each period parallelogram if counted with multiplicities. Furthermore we know that their sum must be equal modulo Λ: Proposition A.15 ([40] II,1.5 Theorem 5). Denote by b1 , b2 , . . . , br the zeros and by a1 , a2 , . . . , ar the poles of an elliptic function f . Then b1 + b2 + · · · + br ≡ a1 + a2 + · · · + ar
mod Zω1 + Zω2 .
Now choose representatives of the congruence classes [ai ] such that b1 + b2 + · · · + br = a1 + a2 + · · · + ar . Then define the function F (u) =
σ(z − b1 )σ(z − b2 ) · · · σ(z − br ) . σ(z − a1 )σ(z − a2 ) · · · σ(z − ar )
From equation A.2.1 we derive for a function ϕ(z) :=
σ(z−a) σ(z−b)
ϕ(z + ω) = exp(η(a − b))ϕ(z), and therefore F (z + ω) = exp(η(a1 − b1 )) + η(a2 − b2 ) + · · · + η(ar − br ). By the choices we made for the ai : F (z + ω) = F (z). So F is a meromorphic function with the periods ω1 and ω2 with its poles at [ai ] and its zeros at [bi ]. Now the quotient of f and F as neither poles nor zeros and is elliptic; it follows from Liouville that it must be a constant. We just have proven the following proposition:
A.2 The meromorphic functions with two periods
163
Proposition A.16. Let f (z) be an elliptic function with b1 , . . . , br a complete system of poles, a1 , . . . , ar a complete system of zeros, such that a1 + a2 + · · · + ar = b1 + b2 + · · · + br . Then there is a constant C ∈ C such that f (z) = C
σ(z − b1 )σ(z − b2 ) · · · σ(z − br ) . σ(z − a1 )σ(z − a2 ) · · · σ(z − ar )
As a last statement we have: Proposition A.17. σ(z) =
ω2 θ10 (0)
exp
¡ ηz 2 ¢ ¡ z ¢ ω2 θ1 ω2 ,
(A.2.2)
where η = η( 12 ω2 ) = ζ( ω22 ). The prime denotes the derivative with respect to z. And for τ = ωω21 θ1 (z) = −i
X
exp(2πiz(n + 12 ) + iπτ (n + 21 )2 + iπn 21 )
n∈Z
=i
X
exp(πiτ (n + 12 )2 + 2πizn + πiz + iπn 12 + 21 πi)
n∈Z
=i =
X
exp(πiτ (n + 12 )2 + 2πi(n + 12 )(z + 21 ))
n∈Z iθ[ 12 , 12 ](τ, z).
164
Theta-Functions
Appendix B
Homological Algebra B.1
Basic definitions
These are just standard definition and can be found in any book on homological algebra, for examplethe books of Gelfand and Manin [28], [29] or the book of Mac Lane [53]. For an introduction to derived categories see Thomas [74], for an introduction to A∞ categories and algebras, we refer to works of Keller [45, 44] Definition B.1. A category C is the datum of • a class of objects, denoted Ob(C), • for each ordered pair of objects A1 , A2 ∈ Ob(C) a set Hom(A1 , A2 ), whose elements are called morphisms or arrows, and • for all objects A1 , A2 , A3 a morphism Hom(A1 , A2 ) × Hom(A2 , A3 ) − → Hom(A1 , A3 ), such that the following axioms are fulfilled: 1. The spaces Hom(A1 , A2 ) are pairwise disjoint. This means a morphism ϕ determines uniquely two objects A1 , A2 , such that ϕ ∈ Hom(A1 , A2 ). 2. There is an identity in each Hom(A, A). 3. The composition is associative. Definition B.2. The opposite category C op of a category C is a category that has the same objects as C has, but with all arrows reversed, that is HomC op (A1 , A2 ) := HomC (A2 , A1 ). Definition B.3. A (covariant) functor F from a category C to a category D is the datum of • a map Ob C − → Ob D, A 7→ F (A), and • a mappings Mor C − → Mor D, Hom(A1 , A2 ) − → Hom(F (A1 ), F (A2 )), ϕ 7→ F (ϕ)
166
Homological Algebra
such that the following equation holds for all morphisms ϕ, ψ in C : F (ϕ ◦ ψ) = F (ϕ) ◦ F (ψ). A contraviarant functor from a category C to the category D is a functor from C op to D. Definition B.4. A category C is called a subcategory of a category D, if Ob C ⊂ Ob D, HomC (A, B) ⊂ HomD (A, B), for all A, B ∈ Ob C and the composition in C equals the composition in D. The subcategory is called full, if HomC (A, B) = HomD (A, B) for all objects in C. A functor is called faithful, if the map on the homomorphism spaces is injective for all objects. It is called full if all these maps are surjective. A functor F : C − → D is called an equivalence of categories, if it is fully faithful and for each object B in D there exists an object A in C, such that F (A) is isomorphic to B. Two categories C, D are equivalent, if there exists an equivalence F : C − → D.
B.2
A∞ -categories with transversal structure
A by now classical reference is Keller’s work [45, 43] on A∞ -algebras. A clear treatment in the light of mirror symmetry can be found in Seidel’s book [67]. The definition of a transversal structure can be found in the paper [62] of Polishchuk. Definition B.5. Let k be a field. A non-unital A∞ - category C consists of the following data: • A set of objects Ob C, • a graded k-vectorspace Hom(X0 , X1 ) for each ordered pair of objects (we denote the degree of an element a ∈ Hom(Xo , X1 ) by |a|), and • composition maps of every order d ≥ 1, µdC : HomC (X0 , X1 ) ⊗ · · · ⊗ Hom(Xd−1 , Xd ) −→ HomC (X0 , Xd )[2 − d],
(B.2.1)
where [k] denotes the usual shift in the grading. Such that the data fulfills the following axioms 1. The morphism spaces HomC (A, B) are pairwise disjoint that means a morphism ϕ determines A, B uniquely such that ϕ ∈ HomC (A, B), and 2. the maps in B.2.1 have to satisfy the following A∞ - relations: X (ad , . . . , an+m+1 , µdC (an+m , . . . , an+1 ), an , . . . , a1 ) = 0, (−1)sn µd−m+1 C m,n
where the sum runs over all possible terms 1 ≤ m ≤ d, 0 ≤ n ≤ d − m and sn = |a1 | + · · · + |an | − n.
B.2 A∞ -categories with transversal structure
167
Remark B.6. Note that an A∞ - category is in general not a category: The composition µ2 need not be associative (in fact, the A∞ operations are built to compensate for the missing associativity) and there might be no identity morphisms contained in Hom(A, A). There are a lot of different possible definitions of A∞ -categories concerning the unit. Some authors require so called weak units to exists. we refer to Seidel [67] and also [68] for the discussion. Definition B.7. An A∞ −category C with transversal structure is the datum of • a class of objects, denoted Ob(C) • distinguished class of (ordered) pairs of objects in Ob(C) × Ob(C). Its elements are called transversal pairs. An ordered n−tupel of objects of C, (A1 , A2 , . . . , An ) is called transversal, if all the pairs (Ai , Aj ), i ≤ j are transversal. • for each transversal pair a set Hom(A1 , A2 ) • for each transversal n-tupel a morphism µn : Hom(A0 , A1 )×Hom(A1 , A2 )×· · ·×Hom(An−1 , An ) − → Hom(A1 , An )[2−n], such that the following axioms are fulfilled: 1. The spaces Hom(A1 , A2 ) are disjoint. 2. The µn fulfill the usual A∞ −relations.
168
Homological Algebra
Appendix C
Intersection Theory We only consider non-singular ambient varieties. In this brief summary we follow mainly the book of Fulton [27]. The general theory and the proofs for everything presented here can be found in this book. The main reference for the section on line bundles is the book of Griffiths and Harris [34]. Definition C.1. Let X be an algebraic scheme. A k-cycle on X is a finite formal sum X ni [Vi ], where the Vi are k-dimensional irreducible subvarieties of X, and ni ∈ Z. For any (k + 1)-dimensional subvariety W of X, and any r, nonzero rational function on W , define a k-cycle [div(r)] on X by X [div(r)] = ordV (r)[V ], (C.0.1) where the sum runs over all codimension one subvarieties V of W and ordV (r) is the order of the function r ∈ R(W )∗ along the subvariety V , defined by the local ring OV,W . The group of k-cycles on X is denoted Zk (X). A k-cycle is rationally equivalent to zero, if there are a finite number of (k + 1)− dimensional subvarieties Wi of X and ri ∈ R(Wi )∗ , such that X α= [div(ri )]. Cycles equivalent to zero form a subgroup of Zk (X), denoted Ratk (x). The group of k-cycles modulo rational equivalence on X is the factor group Ak (X) = Zk (x)/Ratk (X). Define Z(X) (resp. A(X)) to be the direct sum of the Zk (X) (resp. Ak (X)) for k = 0, 1, . . . , dim(X). A cycle (resp. cycle class) on X is an element of Z(X) (resp. A(X)). Definition C.2. Let X be an n-dimensional variety. A (Weil-) divisor on X is an (n − 1)-cycle onPX. A divisor D = ni [Vi ] is called effective, if all ni ≥ 0. A hypersurface V is usually P identified with the divisor [Vi ], where Vi are its irreducible components.
170
C.1
Intersection Theory
Divisors and line bundles on Riemannian manifolds
In this section we treat divisors and line bundles in the special case of a complex manifold. We repeat the basic definitions in this special case: Let V ⊂ X by an irreducible analytic hypersurface of X. For any holomorphic function g defined near p we define the order ordV,p (g) of g along V at p to be the largest integer a such that in the local ring OX,p : g = f a · h. Actually this definition is independent of p, so we simply write ordV (f ). Now let f be a meromorphic function on X with locally f = hg , with g, h holomorphic and relatively prime. Then we define ordV (f ) = ordV (g) − ordV (h). We define the divisor of a meromorphic function f by X (f ) = ordV (f ) · [V ]. V
Such a divisor is called principal. Two divisors D1 , D2 are called linearly equivalent, denoted D1 ∼ D2 if they differ only by a principal divisor (so they are rationally equivalent cycles). On an elliptic curve, we can easily decide, if two divisors are linearly equivalent: Proposition C.3. Given distinct points a1 , . . . , aq contained in C/Λ, and given integers n1 , . . . , nq , a P necessary and sufficient condition that there exists an elliptic function with divisor ni (ai ) is that P 1. ni = 0 and P 2. ni ai = 0 in the group C/Λ Proof. Let xj and yj be points in C such that if ni > 0 there is a j such that xj = ai and if n < 0 there is a j such that ai = yj and such that the number of possible j is ni . With other words, let r(z) be the desired function, than denote by xi its zeroes and by yi its poles. The number of poles and zeros is the same, because we have condition 1. - Denote this number by d. We can choose lifts of the points in C such that d X i=1
xi =
d X
yi ,
(C.1.1)
i=1
not only modulo Λ. Then the function, we are looking for is Qd σ(z − xi ) r(z) = Qi=1 . d i=1 σ(z − yi )
(C.1.2) (C.1.3)
For the details see appendix A.2 or the book of Hurwitz [40] II.2. Divisors can also be described in sheaf theoretic terms. On complex manifolds the two descriptions are equivalent, in general this would be Cartier divisors, which are distinguished from Weil divisors. Denote by MX the sheaf of meromorphic functions on the complex manifold X. Then a (Cartier) divisor is a global section ∗ . It is not difficult to see that these two notions are of the quotient sheaf M∗X /OX
C.1 Divisors and line bundles on Riemannian manifolds
171
∗ is equivalent: On the one hand, a global section {f } of the quotient sheaf M∗X /OX given by the data of an open covering {Uα } of X and meromorphic functions fα on Uα with fα ∗ ∈ OX (Uα ∩ Uβ ), fβ
therefore the order of fα and fβ on a submanifold V of X agree and we can define the (Weil) divisor associated to f by X D(f ) = ordV (fα ) · V, V
where for V we choose Uα , such that V ∩ Uα 6= ∅. On the other hand, given a divisor X D= ai · V i , we choose an open cover {Uα } of X, such that each in each Iα all Vi contributing to D have local defining functions gα,i ∈ O∗ (Uα ). Then we define Y a i , fα := gα,i i ∗ . We established: which defines a global section in M∗X /OX ∗ ∼ H 0 (X, M∗X /OX ) = Div(X).
Recall that holomorphic line bundles on a complex manifold X correspond to ele∗ ) via transition functions: ments in H 1 (X, OX Suppose we are given a holomorphic line bundle L − → X on X. Then we can choose an open covering {Uα } of X and on Uα local trivializations ϕα of L ϕα : L|Uα − → Uα × C. Transition functions gαβ : Uα ∩ Uβ − → C∗ of L relative to the trivializations {(Uα , ϕα )}, are defined by gαβ (z) = (ϕα ◦ ϕβ−1 )|Lz . ∗ ) represented by {g } is independent of the trivialization The element in H 1 (X, OX αβ and defines the line bundle uniquely. With the tensor product of line bundles, ∗ ) gets a group structure; this group is also denoted by Pic(X) and is H 1 (X, OX called the Picard group of X. Given a divisor D with local defining functions fα ∈ M∗X (Uα ), then define
gαβ :
fα . fβ
These are holomorphic transition functions of a line bundle, which we call the line bundle associated to the divisor D, denoted [D]. [·] : Div(X) − → Pic(X) is a functorial isomorphism with many nice properties. For more details we refer to the book of Griffiths and Harris [34]. The correspondence also works in the opposite direction:
172
Intersection Theory
Proposition C.4. Let X be a complex compact manifold. Let D0 be a divisor on X and let L = [D0 ] be the corresponding line bundle. Then 1. for each nonzero s ∈ Γ(X, L), the divisor (s) is an effective divisor linearly equivalent to D0 ; 2. every effective divisor linearly equivalent to D0 is (s) for some s ∈ Γ(X, L); and 3. two sections s, s0 ∈ Γ(X, L) have the same divisor of zeros if and only if there is a λ ∈ k ∗ such that s = λs0 . The proof can be found in Griffiths and Harris [34] or for nonsingular projective varieties over a algebraically closed field in Hartshorne [37] (this is Proposition II.7.7.) Let M be a compact complex manifolds of dimension n. The short exact sequence exp
0− →Z− →O− → O∗ − → 0, gives us a long exact sequence on cohomology. In particular we get a boundary map δ : H 1 (X, O∗ ) − → H 2 (X, Z). For a line bundle L ∈ Pic(X) = H 1 (X, O∗ ) on X we define its Chern class c1 (L) = δ(L). For a divisor we define its Chern class as the Chern class of the corresponding line bundle. Proposition C.5. Two holomorphic line bundles with the same Chern class differ by an C ∞ isomorphism. As all results in this section this can be found in Griffiths and Harris [34]. On a compact connected Riemann surface H 2 (X, Z) ∼ = Z, the isomorphism is given by integrating a form over X. Throughout this work we have used this identification. Sometimes this is used as a definition of the degree of a line bundle. Let L be the P line bundle corresponding to the divisor D = ai · [Vi ]. we have defined the degree of the line bundle L as the P degree of its corresponding divisor, and the degree of D was defined by deg D = ai . These definitions match, because c1 ([D]) = deg D. We define the Chern class of a divisor as the Chern class of the corresponding line bundle. Remark C.6. Note that for a compact connected Riemann surface X the Chern class and degree agree under the isomorphism H 2 (X, Z) ∼ = Z given by the natural orientation on X as is shown in Griffiths and Harris [34, chapter 1.1.].
C.2 Intersections
C.2
173
Intersections
Definition C.7. (Proper pushforward of cycles) Let f : X − → Y be a proper surjective morphism, with X, Y complex varieties. For any subvariety V of X, the image W = f (V ) is a subvariety of Y . If W has the same dimension als V , f defines a finite covering with welldefined number of sheets nV /W . Define the degree of V over W by ( nV /W if dim(V ) = dim(W ) deg(V /W ) = 0 if dim(V ) > dim(W ). Define f∗ (V ) = deg(V /W )[W ]. This extends linearly to a functorial homomorphism f∗ : Zk X − → Zk Y, called pushforward along f . The pushforward respects cycle classes, so we get a welldefined, functorial homomorphism: f∗ : Ak (X) − → Ak (Y ). Definition C.8. (Flat pullback of cycles) Let f : X − → Y be a flat morphism of relative dimension n (our most important example is the projection from a Cartesian square X × Y − → X, with Y irreducible and n - dimensional). Then define the pullback along f by f ∗ [V ] = [f −1 (V )], for a irreducible subvariety V ⊂ Y . This extends by linearity to a homomorphism f ∗ Zk (Y ) − → Zk+n (X), which is functorial. It respects the cycle classes and extends therefore to a homomorphism f ∗ : Ak (Y ) − → Ak+n (X). Definition C.9. (Intersection product of cycles) On Y , a non-singular variety, one can define an intersection product Ak (Y ) × Al (Y ) − → Ak+l−n (Y ), where n is the dimension of Y . For V ∈ Ak (Y ) and W ∈ Al (Y ) it is denoted by V ·W. Given two nonsingular, irreducible varieties V ∈ Ak (Y ), W ∈ An−k (Y ) that meet transversally, then the intersection product is the usual set theoretic intersection. Remark C.10. The whole book of Fulton [27] is dedicated to more and more general definitions of this intersection product. We will not need a more general definition here, and only refer to [27] for the definition.
174
Intersection Theory
Definition C.11. A generalized correspondence from a (nonsingular) variety X to a (nonsingular) variety Y , is a cycle or cycle class on X ×Y. We denote a (generalized) correspondence α between X and Y by α : X ` Y . If α : X ` Y, β : Y ` Z, we define the composite of α and β by the formula β ◦ α = pXZ (p∗XY α · p∗Y Z β)
with
X × Y ×NZ p
pXY pppp
ppp px pp
X ×Y
²
NNN NpNNXZ NNN N&
pY Z
y×Z
X ×Z
Note that β ◦ α is in general defined only up to rational equivalence (one may have to choose a different representative of the cycle class, such that p∗XY α, p∗Y Z β meet properly). A (generalized) correspondence α : X ` Y has a transpose α0 : Y ` X defined by α0 = r∗ (α), where r : X × Y − → Y × X reverses the factors. An irreducible (generalized) correspondence from X to Y is an irreducible subvariety V of X × Y, identified with its cycle. Example C.12. Any morphism f : X − → Y determines an irreducible correspondence Γf from X to Y, given by the graph embedding of X in X × Y. Proposition C.13. ( First properties) Let α : X ` Y, β : Y ` Z. 1. The composition of (generalized) correspondences is associative: If γ : Z ` W , then γ ◦ (β ◦ α) = (γ ◦ β) ◦ α. 2. The transposition of (generalized) correspondences: (β ◦ α)0 = α0 ◦ β 0 and (α0 )0 = α. 3. Graphs: If α = Γf and β = Γg , then β ◦ α = Γg◦f . Remark C.14. The product α × β − → α ◦ β makes A(X × X) into an associative ring with unit [∆X ], and with involution α − → α0 Definition C.15. For α : X ` Y, define a homomorphism α∗ : A(X) − → A(Y ) XY ∗ (a) · α), and a homomorphism by the formula α∗ (a) = pXY Y ∗ (pX
α∗ : A(Y ) − → A(X) XY ∗ (b) · α). by the formula α∗ (b) = pXY X∗ (pY
Remark C.16. In section 8 we define (in full analogy) for such a (generalized) correspondence α with structure sheaf Oα the functors: α∗ : Db (X) − → Db (Y ) L
XY ∗ (a) ⊗ O ), and a functor by the formula α∗ (a) = RpXY α Y ∗ (pX
α∗ : Db (Y ) − → Db (X) L
XY ∗ (b) ⊗ O ). by the formula α∗ (b) = RpXY α X∗ (pY
C.2 Intersections
175
The following proposition is true for the induced map between the cycles of X and the cycles of Y , as well as for the induced functors between the derived categories of coherent sheaves. Proposition C.17. (Properties of the pullback and pushforward along (generalized) correspondences) 1. If α : X ` Y, β : Y ` Z, then (β ◦ α)∗ = β∗ ◦ α∗ and (β ◦ α)∗ = α∗ ◦ β ∗ . 2. If α : X ` Y, then (α0 )∗ = α∗ . 3. If f : X −→ Y, then (Γf )∗ = f∗ and (Γf )∗ = f ∗ . Definition C.18. Let C(X × Y ) ∈ An (X × Y ), with n = dimX. Note that this subset is closed under transposition. We will call this the set of correspondences. Remark C.19. In our case X and Y will be elliptic curves, so correspondences are divisor classes in X × Y. Remark C.20. Several modifications of A(X × Y ) are used in the literature, and are all called correspondence. Definition and Proposition C.21. Let I(X, Y ) be the subgroup of A(X × Y ) generated by correspondences of the form [V × W ], with V (resp. W ) a subvariety of X (resp. Y ); such correspondences are called degenerate. IF α ∈ I(X, Y ) and β ∈ I(Y, Z), then β ◦ α ∈ I(X, Z). (It follows that I(X, X) is a two sided ideal in A(X × X), closed under transposition.) Definition and Proposition C.22. Let X, Y, Z be of the same dimension n. The degrees d1 (α) and d2 (α) of a correspondence α ∈ C(X × Y ) are defined by XY PX∗ (α) = d1 (α) · [X],
pXY Y∗ (α) = d2 (α) · [Y ].
1. If β ∈ C(Y × Z), then for i = 1, 2: di (β ◦ α) = di (β) · di (α). 2. For all α ∈ C(X × Y ) : d1 (α0 ) = d2 (α). With a = d1 (α) and b = d2 (α), α is called an (a, b)−correspondence.
176
Intersection Theory
Appendix D
Not yet a mirror functor for families of elliptic curves Other than the previous appendices this appendix does not only contain material from textbooks. In this chapter we focus on a natural continuation of our problem before. We constructed in chapter 3 the 2-category of the derived categories of elliptic curves for the tautological family E − → H of elliptic curves with a choice of a basis, that means the fibre at τ is Eτ = (Eτ , τ ). We now ask, how we can built a mirror functor for more general families of elliptic curves and what categories we should use for this task. The construction of such a mirror symmetry exceeds, however, this thesis by far. We merely report on some preliminary thoughts on this.
D.1
Definitions
This is a brief recollection of the definitions we use. For a coherent introduction see for example Shimizu and Ueno [72] for complex analytic theory, Hartshorne [37], Demazure and Gabriel [18] or Mumford [56] for the algebraic theory. Definition D.1. Suppose given a n- dimensional connected compact complex manifold Mt for each point t of a domain B of Rm . The set {Mt |t ∈ B} is called a smooth family or differentiable family of compact complex manifolds, if there are a differentiable manifold M and a C ∞ -map ω of M into B satisfying the following conditions: 1. The rank of the Jacobian of ω is equal to m at every point of M. 2. For each t ∈ B, ω −1 (t) is a compact connected subset of M. 3. ω −1 (t) ∼ = Mt . 4. There are a locally finite open covering {Uj |j ∈ I} of M with I a finite index set and complex valued C ∞ functions zj1 (p), . . . , zjn (p), j ∈ I defined on Uj such that for each t ¡ ¢ {p 7→ zj1 (p), . . . , zjn (p) |Uj ∩ ω −1 (t) 6= ∅} form a system of local complex coordinates on Mt .
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Remark D.2. instead of B ∈ Rm we can of course choose B a domain in a mdimensional differentiable manifold (B is in that case isomorphic to a domain in Rn ). Definition D.3. Let B be a domain in Cm . A set {Mt |t ∈ B} of connected complex manifolds Mt , depending on t = (t1 , . . . , tm ) ∈ B. We say that Mt depends holomorphically on t and that {Mt |t ∈ B} is a complex analytic family if there is a complex manifold M and a holomorphic map ω of M onto B satisfying the following conditions: 1. The rank of the Jacobian of ω is equal to m at every point of M. 2. For each t ∈ B, ω −1 (t) is a compact complex submanifold of M. 3. ω −1 (t) ∼ = Mt . We can then choose a system of local complex coordinates (z1 , . . . , zj , . . . ), zj : p 7→ zj (p), and coordinate polydiscs Uj with respect to zj such that the following holds true 1. zj (p) = (zj1 (p), . . . , zjn (p), t1 , . . . , tm ) with ω(p) = (t1 , . . . , tm ) 2. U = {Uj } is locally finite. Then
¡ ¢ {p 7→ zj1 (p), . . . , zjn (p) |Uj ∩ Mt 6= ∅}
gives a system of local complex coordinates on Mt . Remark D.4. Instead of B ⊂ Cm , we could of course choose B ⊂ X, for a complex manifold X. A family of elliptic curves, is not a simple family of complex manifolds. We want also the group structure to vary holomorphically. Here is the definition: Definition D.5. Let S be a complex manifold. A family of elliptic curves parameterized by S is a holomorphic map E − → S of complex manifolds together with a holomorphic section s : S − → E, such that the following holds: ¡ ¢ 1. all fibres Es , e(s) , s ∈ S are elliptic curves (with neutral element e = e(s)), S 2. there exists an open covering S = i∈I Ui , such that there are holomorphic maps ω1 , ω2 : Ui − → C, with ω1 (z), ω2 (z) are R−linear independent for all z and such that Es ∼ = C/(ω1 (s)Z + ω2 (z)Z) and such that the section e is given by e|Ui : Ui − → π −1 (Ui ) s 7→ (0, s) A family of elliptic curves E parameterized by S is also called elliptic curve over S or relative elliptic curve E/S.
D.1 Definitions
179
Example D.6. Let S = H, E = C × H/(z, τ ) ∼ (z + nτ + m, τ ), ∀n, m ∈ Z, and E− → H, [z, τ ] 7→ τ. This is called the universal or tautological family over H. There are other examples of families of elliptic curves, for example the Weierstrass family or the Legendre family for more examples and details see the book of Katz and Mazur [42]. Definition D.7. Let F be a contravariant functor from the category of complex spaces to sets. A fine moduli space (M, µ) for F is a complex manifold F together µ with an isomorphism of functors F ∼ = Hom(·, F). A coarse moduli space for F is a complex manifold M together with a natural transformation of functors F − → Hom(·, M), such that for a point p the morphism F(p) − → Hom(p, M) ∼ = M is bijective. Theorem D.8. Via the family E −→ H, the space H is a fine moduli space of elliptic curves E, together with the choice of an oriented basis, v, w ∈ C, such that E∼ = C/(Zv + Zw). Proof. This is cited from Friedman and Morgan [21] theorem 3.7. Theorem D.9. H/ PSL2 (Z) together with the tautological family is a coarse moduli space for elliptic curves over C. That means for each family E˜ −→ S of complex elliptic curves there is a unique holomorphic map f : S −→ H/P SL2 (Z) with E˜s ∼ = Ef (s) ∀s ∈ S. The proof can be found in Mumford [56]. Proposition D.10. Let S be an elliptic surface, i.e. S is a complex surface with a holomorphic map π : S −→ C, where C is a curve, such that for a general point t ∈ C, the fibre π −1 (t) is a smooth curve of genus one. Then the following are equivalent: 1. There exists a holomorphic line bundle L on S, such that L|f has positive degree, where f is any irreducible fibre of π 2. S is algebraic 3. There exists a multisection X of π, i.e. an irreducible curve X on S such that X is not contained in a fibre f of π, or, equivalently, such that X · f ≥ 1. In particular, every elliptic surface with a section is algebraic. Following Grotendieck cite[§20.21]egaiviv we can define relative meromorphic functions and then analogously to the procedure one defines Cartier divisors from the sheaf of meromorphic functions we get the notion of relative Cartier divisor form the definition of relative meromorphic functions. The following theorem suffices for our purposes in lieu of a definition
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Theorem D.11.
Div1 (E/S) ∼ = {sections s : S −→ E}
And we get all devisors by formal sums from divisors of degree one.
D.2
Preliminary thoughts
In chapter 3, we already have constructed an equivalence Φ : pDb ellall − → pFKellall , between the 2-category of pairs of a derived category of a complex torus and a choice of a base of the lattice, and all functors between them as morphisms, resp. the 2-category of pairs of a Fukaya category of a real 2-torus with the choice of a complexified symplectic form, and all functors as morphisms. We interpreted this as a trivial example of an extension of the mirror functor to the elliptic curve E − → H. We can repeat this trivial result for all families of elliptic curves: The trivial example Let X − → S be a smooth family of elliptic curves over C. We define the 2-category pDb X/S all of elliptic curves over S (again, p stands for pairs and all for all functors) as follows: objects are {(Db (Xs ), s)|s ∈ S} ¡ ¢ morphisms are given by Hom (Db (Xs ), s), (Db (Xs0 ), s0 ) © ¡ ¢ª = triples (s, s0 , F ) with F ∈ Funct Db (Es ), Db (Es0 ) According to theorem D.8 there is a unique map τˆ : S − → H/P SL2 (Z) such that for each s ∈ S the fibre Xs is isomorphic to Eτˆ(s) . Remark D.12. The quotient E/ PSL2 (Z) does not exist in the category of complex manifolds. Therefore we cannot get a fine moduli space for elliptic curves like this. We choose therefore a non-unique lift of τˆ s9 H ss
τ sss
ss ss ² s s S τˆ / H/ PSL2 (Z)
such that still is : Xs ∼ = Eτ (s) for all s ∈ S. Having such isomorphisms fixed for each s ∈ S we now define a functor which we also denote by τ : τ : pDb X/S all − → pDb ellall by assigning: ¡ ¢ (Db (Xs ), s) 7→ Db (Eτ (s) ), τ (s) ¡ ¢ ¡ ¢ Hom (Db (Xs ), s), (Db (Xs0 ), s0 ) − → Hom (Db (Eτ (s) ), τ (s)), (Db (Eτ (s0 ) ), τ (s0 )) F 7→ is∗ ◦ F ◦ i∗s0 .
D.2 Preliminary thoughts
181
Note that, since is is an isomorphism of each s, is∗ and i∗s are inverse equivalences. The functor τ is fully faithful, and an equivalence if τ (S) = H. We can now compose τ with the previously defined functor Φ to get a fully faithful functor Φ ◦ τ : pDb X/S all − → pFKellall . With all these choices for the smooth family X − → S, we can define a mirror family ˆ X− → S by choosing the mirror torus separately in each fibre. (Note that we need the choice of τ : s − → H here.) If we restrict the category pFKellall to the subcategory, whose objects are (FK(Et ), t) with t in the image of τ , then this is an equivalence. So we established the equivalence: Φ ◦ τ : pDb X/S all − → pFKell/S all . Remark D.13. We made several choices: First we chose a lift of S − → H/P SL2 (Z), and then we chose for each fibre of our family an isomorphism is . In both cases, different choices differ in each fibre by an automorphism of the elliptic curve. In the previous chapter we have investigated at length, how automorphisms change the mirror functor Φ, which was defined in each fibre separately, and depends as well on these choices. Of course this is again a trivial example that gives us no new insights, and as before we conclude that we need a different category. While before, we chose to restrict the morphisms to those with a geometric meaning, we now are looking for a way to built a new category with new objects. Instead of performing a mirror symmetry fibrewise we would now like to construct global objects on X − → S, that restrict to elements of the derived category fibrewise and perform mirror symmetry for these global objects. The general idea is the following: Roughly speaking, mirror symmetry states that deformations of the complex structure correspond to deformations of the symplectic structure on the mirror manifold. Over these deformation of elliptic curves (or a family of elliptic curves, deformations are infinitesimal families, see above) we have fibrewise our derived category, and it would be nice if we had a kind of connection that provides us for each element of the derived category in one fibre, with its corresponding deformation in the other fibre. These deformations of coherent sheaves, or let us first only think of deformations of vectorbundles, may form these global object. Before we go deeper into this, we do some preliminary calculations that show us the limit of such constructions if we want to work globally. Globally here means, we consider not deformations of elliptic curves and vectorbundles, but with families of elliptic curves over an arbitrary base and vectorbundles on the total space of our family. The failure of the hands-on approach for a connection on the derived categories: Why we cannot get a 1-1 correspondence of the objects of two categories over two different fibres Consider the following problem: Given a smooth family {Ek } of elliptic curves over C, construct a map that allows us to compare objects and morphisms in the categories over different fibres and tell, which object is deformed into which other object over a different fibre. We would love this map to be functorial, at least for
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isomorphic elliptic curves and therefore equivalent derived categories, we would like the resulting map to be functorial. If we can solve this problem and construct such a functor (or map), we would call it a connection on our family of derived categories of the family of elliptic curves. Suppose for the family above, τ (k) gives us a path in H connecting two different points τ and τ 0 . For this path we choose a parametrization p : [0, 1] − → H. Now by substituting τ by p(t) in all formulas and identifying Lτ with Lp(t) , θ(τ, z) with θ(p(t), z) and so on, finally we define a map Db (Eτ ) − → Db (Eτ 0 ). But Proposition D.14. The map between categories Db (Ep(t1 ) ) −→ Db (Eτ2 ), induced by replacing everywhere p(t1 ) with p(t2 ) is not functorial in general. Proof. Theorem 7.1 already stated that in the case p(t1 ) = τ, p(t2 ) = τ + 1, this does not yield a functor. For to see that this is not the only exception to functoriality, we look at another example: Example D.15. We give a counterexample with p(t1 ) = τ, p(t2 ) = − τ1 . Consider three line bundles L(ϕi ) on Eτ and their counterparts L(ψi ) on E 1 with ϕi = tx∗ i ϕ0 ϕn0 i −1 , xi ψi = t∗yi ψ0 ψ0ni −1 , yi =
−τ 1 = ai τ + bi and n3 > n2 > n1 and ψ0 = ϕ0 (− τ , z) and y −y x −x ai (− τ1 ) + bi . Define nji = nj − ni and xji = njj −nii , yji = njj −nii ,
for the morphisms spaces we then have the bases: {fkji = t∗xji θ[ nkji , 0](nji τ, nji z)} {gkji = t∗yji θ[ nkji , 0](nji (− τ1 ), nji z)}
¡ ¢ for Hom L(ϕi ), L(ϕj ) ¡ ¢ for Hom L(ψi ), L(ψj ) .
Under the map we are investigating, the morphism spaces are transformed into each other by mapping fkji to gkji . Now we check the functoriality, using formula A.5 ¡ ¢ ¡ ¢ fa21 · fb32 = θ[ na21 , 0] n21 τ, n21 (z + x21 ) θ[ nb32 , 0] n32 τ, n32 (z + x32 ) X ¡ ¢ d+n32 a+n21 b θ[ n21 n32 , 0] n21 n32 n31 τ, n21 n32 (x21 − x32 ) = n21 n32 n31 ¡ ¢ d∈Z/n31 Z ·θ[ n21 nd+a+b , 0] n31 τ, n31 (z + x31 ) 31 X = Ck fk31 , d∈Z/n31 Z
with k = n21 d + a + b. The Ck are elements of C and factors in front of the basis vectors fk31 . So this is mapped to fa21 fb32 =
X d∈Z/n31 Z
=
X
Ck fk31 7→
X d∈Z/n31 Z
Ck gk31
¡ ¢ d+n3 2a+n21 b θ[ n21 n32 , 0] n21 n32 n31 τ, n21 n32 (x21 − x32 ) n21 n32 n31 ¡ ¢ d∈Z/n31 Z θ[ n21 nd+a+b , 0] n31 (− τ1 ), n31 (z + y31 ) . 31
D.2 Preliminary thoughts
183
Note: xji 6= yji . On the other hand ¡ ¢ ¡ ¢ ga21 · gb32 = θ[ na21 , 0] n21 (− τ1 ), n21 (z + y21 ) θ[ nb32 , 0] n32 (− τ1 ), n32 (z + y32 ) X ¡ ¢ d+n32 a+n21 b = θ[ n21 n32 , 0] n21 n32 n31 (− τ1 ), n21 n32 (y21 − y32 ) n21 n32 n31 ¡ ¢ d∈Z/n31 Z , 0] n31 (− τ1 ), n31 (z + y31 ) θ[ n21 nd+a+b 31 X = Dk gk31 , d∈Z/n31 Z
with k = n21 d + a + b. For our map to be functorial we therefore need Dk = Ck for all k: ¡ ¢ d+n32 a+n21 b Dk = θ[ n21 n32 , 0] n21 n32 n31 (− τ1 ), n21 n32 (y21 − y32 ) n21 n32 n31 ¡ ¢ d+n32 a+n21 b Ck = θ[ n21 n32 , 0] n21 n32 n31 τ, n21 n32 (x21 − x32 ) n21 n32 n31 But these two are not equal. We take a slightly different view on this problem of constructing a parallel transport of objects: Together with a path p(t) in H and the smooth family we get from the restriction of the tautological elliptic curve to this path, we consider a smooth family of divisors of degree 1 that restricts to the point aτ + b in the fibre over τ = p(0). This is the same as a smooth section in our family. It corresponds furthermore to a smooth family of line bundles with the line bundle Lτ (t∗x ϕ0 ), x = − 12 τ − 12 + aτ + b fixed over Eτ . Obviously D0 is such a family of divisors: n o D0 := (p(t), ap(t) + b) ⊂ (H × C)/SL2 (Z) o Z = E. The corresponding family of line bundles is: ¡ ¢ ¢ ¡ ¢ ¡ ¢ ¡ L(D0 ) = C × (H × C)/ v, p(t), z ∼ v, p(t), z + 1 ∼ t∗x ϕ0 z, p(t) , p(t), z + p(t) , where ϕ0 (z, p(t)) = exp(−πip(t)−2πiz), x = − 21 τ − 21 +aτ +b. And what we showed above was, that the map L(D0 )|t0 7→ L(D0 )|t1 , is not functorial. Now take any lift µ ¶ a(t) b(t) M (t) = c(t) d(t) of p(t) to SL2 (R). Such a lift is a path M (t) ⊂ SL2 (R), such that M (t)τ = p(t), M (0) = 1. The condition gives us two equations (for the real- und the imaginary part) in 4 unknowns. The second condition prescribes the connected component of SL2 (R), we could also ask for M (0) = −1. Now define for a divisor of degree one p0 ∈ Eτ a family corresponding to M (t) by: n³ ´o ¡ ¢ p0 D p0 , M (t) := M (t)τ, . c(t)τ + d(t) This definition formally extends to divisors of arbitrary degree. We still have some freedom for the choice of M (t), so that we can pose some conditions. For example we could require that if we know that for t1 , t2 we have:p(t1 ) =
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Not yet a mirror functor for families of elliptic curves
τ, p(t2 ) = γ · τ , that then M (t2 )M (t1 )−1 = γ. This would avoid the counterexamples we have constructed in this specific example, but not in general, since we have to explicitly ask for such a relation for explicit parameters t - there is not a general way to express that a path contains elements of the same orbit of the SL2 (Z) - action on H. But all these conditions cannot fix such a path, which would then be our desired connection. This means we need not a smooth section into the family, but a holomorphic section. This also means we have to consider from the start holomorphic families, and holomorphic sections into holomorphic families E/S are the same as relative divisors of the family (theorem D.11). This means also, we have to give up to look at the family E − → H, and try to find a section that respects the SL2 (Z) - action, because there is no such section (see proposition D.10), because the quotient E/ SL2 (Z) is not algebraic. If we still want to work over H we should switch to deformations instead of families.
List of Symbols (·)∨ A, B, C, . . . α
the dual, sheaves (generally denoted by script letters), a choice of a logarithm of the slope of a Lagrangian Λ in E τ , which gives a grading on the objects of FK(E τ ),
α
a correspondence,
b
b : Eτ − → Eτ +1 is the base change, given by z 7→ z,
C∞ C C(A1 , A2 )
the set of real smooth functions, the field of complex numbers, correction factor for morphisms from A1 − → A2 under Φτ , the category of coherent sheaves on X,
Coh(X) Dτ (ϕ) ∆ Db (X) deg det FK(Y )
Eτ Eτ
the divisor, corresponding to the line bundle Lτ (ϕ), the element of V ⊗V ∨ that corresponds to the identity on V , the derived category of coherent sheaves on X, the degree; used for morphisms and line bundles, the determinant, used for matrices and also for vector bundles, the Fukaya category of the complexified symplectic manifold Y , the standard torus R2 /Z2 = C/(Zi + Z) with the complexified symplectic form τ dx ∧ dy., the Riemann surface C/Λτ , where Λτ is the lattice spanned by τ and 1,
186
List of Symbols
Fτ (V, mτ , m1 ) FA Fα (f, γ)
the vector bundle over Eτ with fibre V and multipliers mτ and m1 , functor Db (Eτ ) − → Db (Eτ ) given by tensoring with the coherent sheaf A, 0 the functor Db (E τ ) − → Db (E τ ), ¡given ¢by the correspondence α by Fα (·) = p1∗ p∗2 (·) , 0 a graded morphism with f : E τ − → Eτ a complex linear symplectic map and γ ∈ g + Z, where πig is the amount by which f turns the imaginary axis,
γs Γ(X, A)
the morphism Eτ − → Esτ given by z 7→ sz, the space of sections of the coherent sheaf A over X if the base space is clear from the context, we also write Γ(A),
H 0 (F )
for F a vector bundle over X, with sheaf of sections denoted by F it denotes the 0-th cohomology H 0 (X, F), which is isomorphic to Γ(X, F ), the global sections of F over X, a short notation for H 0 (X, A), isomorphic the space of global sections of the coherent sheaf A on X, the sheaf of local homomorphisms or sheaf hom, given by U 7→ Hom(A(U ), B(U )),
H 0 (A)
Hom(A, B)
j
the isomorphism Erτ × Z/dZ − → Er1 τ ×Eτ Er2 τ , with r1 , r2 ∈ Z, d = gcd(r1 , r2 ), r = r1 r2 d ,
Lτ (ϕ)
the line bundle over Eτ with multipliers mτ = ϕ and one, a special Lagrangian in E τ , a two dimensional lattice in C,
Λ Λ M m mγ
n νϕ,N
local system on a Lagrangian Λ of E τ , with fibre M , m : Eτ − → E− 1 is given by z 7→ τz , τ the multiplier of a vectorbundle on a space X in direction of γ ∈ Π1 (X), n : Eτ − → E τ given by (x, y)¡7→ (−y, ¢ x), 0 the¡ isomorphism νϕ,N : H → ¢ Lτ (ϕ) ⊗ V − H 0 Lτ (ϕ) ⊗ Fτ (V, exp N ) ,
List of Symbols
OX
pDb ell
pDb ellall pFKell
pFKellall Φτ ϕ0 (τ, z) ΦA
ΦF
πr pr
187
the structure sheaf of X; in the case where X is a complex space, the sheaf of holomorphic functions, ¡ ¢ the category of pairs τ, Db (Eτ ) where the morphism are functors with immediate geometric meaning - pushforwards and pullbacks of morphisms mainly, the category of pairs (τ, Db (Eτ )) with morphisms defined by functors, ¡ ¢ the category of pairs τ, FK(E τ ) where morphisms are given by functors of immediate geometric meaning - pushforwards and pullbacks along morphisms mainly, the category of pairs (τ, FK(E τ )) with morphisms defined by functors, the mirror functor Db (Eτ ) − → F K(E τ ), the function H × C − → C, with ϕ0 (τ, z) = exp(−πiτ − 2πiz), the functor Db (X) − → Db (Y ) induced by the ¡ L object¢ A on X × Y , defined by Rpy∗ A ⊗ p∗X (·) , for a functor F between derived categories of elliptic curves the symplectic counterpart, defined by using the mirror functor twice with respect to the respective elliptic curves, the map Erτ − → Eτ given be the inclusion of lattices, the symplectic map E rτ − → E τ , given by (x, y) 7→ (rx, y),
R r10 , r20 , r
the field of real numbers, for integers r1 , r2 , we always denote by r their least common multiple and by ri0 the quotients ri gcd(r1 ,r2 ) ,
σ
the sigma function, an entire function on C with prescribed zeroes, for integers s1 , s2 , we always denote by s their least common multiple and by s0i the quotients si gcd(s1 ,s2 ) , the special linear group of 2x2 matrices with entries in Z and determinant 1,
s01 , s02 , s
SL2 (Z)
188
List of Symbols
S(x, V, N )
torsion sheaf on Eτ with support at x, fibre at x is V and action of maximal ideal given by the action of N on V ,
τ tx , t∗x
an element of the upper complex half plane, tx is the translation on Eτ by x: tx (z) = z +x, more often we use the pullback with respect to the translation t∗x ϕ0 (z) = ϕ0 (z + x),
Z
the ring of integers,
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Selbst¨ andigkeitserkl¨ arung Hiermit erkl¨are ich, die vorliegende Dissertation selbst¨andig und ohne unzul¨assige Hilfe angefertigt zu haben. Ich habe keine anderen als die angef¨ uhrten Quellen und Hilfsmittel benutzt und s¨amtliche Textstellen, die w¨ortlich oder sinngem¨aß aus ver¨offentlichten oder unver¨offentlichten Schriften entnommen wurden, und alle Angaben, die auf m¨ undlichen Ausk¨ unften beruhen, als solche kenntlich gemacht. Ebenfalls sind alle von anderen Personen bereitgestellten Materialien oder erbrachten Dienstleistungen als solche gekennzeichnet.
Leipzig, 15. Februar 2007
(Hilke Reiter)
Wissenschaftlicher Werdegang der Autorin • 1995 Abitur am Mariengymnasium Warendorf • 1995-1996 Studium Physik (Diplom) an der Georg-August-Universit¨ at zu G¨ottingen • 1996-2001 Studium ev. Theologie und Mathematik f¨ ur das Lehramt an Gymnasien an der Georg-August-Universi¨ at zu G¨ottingen • 2001 Examensarbeit ”Evoluten von ebenen algebraischen Kurven und verwandte Begriffe” bei Prof. Fabrizio Catanese (Note: sehr gut) • 2001 Erstes Staatsexamen in den F¨achern ev. Theologie und Mathematik (Note: 1,1) mit der Gesamtnote: sehr gut (1,2) • 2001-2003 Studienreferendarin am Studienseminar I, Kassel • 2003-2006 Stipendiatin im Graduiertenkolleg ”Analysis, Geometrie und ihre Verbindung zu den Naturwissenschaften” at Leipzig • 2003-2007 Promotionsstudium an der Universi¨
Bibliographische Daten Reiter, Hilke On Functoriality of Homological Mirror Symmetry Dissertation, Universit¨at Leipzig 2007 194 Seiten, 2 Abbildungen, 78 Literaturangaben Mathematics Subject Classification (2000):14J32, 14H52, (18E30, 53D12) Keywords: mirror symmetry, elliptic curve, Fukaya category, Lagrangian submanifolds, derived category
