The University of Hong Kong Department of Mathematics
Mathematics Project Final Report
Topics in Riemann Surfaces and Complex Manifolds
Student Zhong Xingxin Supervisor Prof.Mok Ngaiming
May 27, 2011
Abstract This report is the result of a year long independent study of Riemann surfaces and some topics in complex manifolds. The report can be divided into four parts. From Chapter 1 to Chapter 5, we go over the preliminary definitions and concepts related to Riemann surfaces, including some models of Riemann surfaces (Chapter 1), some topological properties of Riemann surfaces (Chapter 2), concepts and examples of sheaves and cohomology (Chapter 3), definitions of vector bundles, sections and divisors (Chapter 4) which we be used from time to time as we go deeper into the report, and differential and integral on Riemann surfaces (Chapter 5) as a preparation for the study of meromorphic functions on Riemann surfaces. Chapter 6 will be focusing on meromorphic functions on compact Riemann surfaces. As a matter of fact, compact Riemann surfaces are of most interest to us. Due to uniformization theorem, we are able to classify compact Riemann surfaces according to their genera and in the three cases, namely when genus is 0, 1 and at least 2, we view compact Riemann surfaces as quotients of group actions and use different methods, namely elliptic functions and Poincar´e series, to construct non-constant meromorphic functions on them. In Chapter 8, we will see, however, the methods we adopted in the previous chapter cannot be generalized to higher dimensional compact complex manifolds, among which complex torus will be our focus. We will be concerned with the question when does a complex torus have non-constant meromorphic functions on it and how to construct them if they exist. In higher dimensional cases, we will not be able to mimic the method we used before because set of poles of multiply periodic meromorphic functions will not break up naturally as a union of local objects, so that the series we would like to construct could not converge. Instead, we will show that any multiply periodic function can be written as the quotient of two theta functions and thus our task can be reduced to construction of theta functions. In Chapter 9 and 10, we will introduce Hermitian and K¨ahler manifold and with focus on the curvature form associated to holomorphic line bundles on Hermitian manifolds. It leads us to the Kodaira embedding theorem, which states a condition
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on the existence of positive line bundles over a compact complex manifold under which the manifold is embeddable into the projective space PN . We will apply the theorem to complex Riemann surfaces to give a different proof from the one we give in Chapter 6, showing the embeddability of compact Riemann surfaces. Chapter 7, which focuses on Abel map and Jacobi variety, is in some sense isolated from the main stream of the report. However, It can be a direction for our future study.
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Acknowledgments I would like to thank Professor Mok Ngaiming, my academic supervisor, for guidance and assistance in every aspect of the project. I would also like to thank Department of Mathematics, HKU, for providing me this opportunity.
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Contents
Abstract
3
Acknowledgments
5
1 Introduction to Riemann Surfaces
11
1.1
Definition of Riemann Surfaces . . . . . . . . . . . . . . . . . . . . .
11
1.2
Non-singular Algebraic Curves . . . . . . . . . . . . . . . . . . . . . .
12
1.3
Quotients under Group Actions . . . . . . . . . . . . . . . . . . . . .
15
2 Topology of Riemann Surfaces
17
2.1
Countable Topology
. . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2
Compact Riemann Surface as Sphere with Handles . . . . . . . . . .
20
2.3
Fundamental Groups . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.4
First Homology Groups . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3 Sheaves and Cohomology
25
3.1
Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.2
Cohomology on Sheaves . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.3
Examples and Applications
31
. . . . . . . . . . . . . . . . . . . . . . . 7
4 Vector Bundles, Sections and Divisors
37
4.1
Vector Bundles and Sections . . . . . . . . . . . . . . . . . . . . . . .
37
4.2
Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5 Differential and Integral on Riemann Surfaces
43
5.1
Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
5.2
Integration of Differential Forms . . . . . . . . . . . . . . . . . . . . .
47
5.3
Abelian differentials of three kinds . . . . . . . . . . . . . . . . . . . .
48
6 Meromorphic Functions on Compact Riemann Surfaces
53
6.1
The Riemann-Roch Theorem and Applications . . . . . . . . . . . . .
54
6.2
Riemann-Hurwitz Formula . . . . . . . . . . . . . . . . . . . . . . . .
57
6.3
Doubly Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . .
60
6.4
Poincar´e Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
6.5
Projective Embedding of Compact Riemann Surfaces . . . . . . . . .
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7 Abel Map and Jacobi Variety
71
7.1
The Jacobian and Abel’s Theorem
. . . . . . . . . . . . . . . . . . .
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7.2
Jacobi Inversion Problem . . . . . . . . . . . . . . . . . . . . . . . . .
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8 Theta Functions
79
8.1
Reduction to Theta Functions . . . . . . . . . . . . . . . . . . . . . .
79
8.2
Theta Functions and Riemann Forms . . . . . . . . . . . . . . . . . .
83
8.3
Construction of Theta Functions . . . . . . . . . . . . . . . . . . . . .
89
9 Hermitian and K¨ ahler Manifolds
99 8
9.1
Hermitian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2
Hermitian Connections and Curvature Forms . . . . . . . . . . . . . . 103
10 Kodaira Embedding Theorem
99
109
10.1 Kodaira-Nakano Vanishing Theorem . . . . . . . . . . . . . . . . . . 109 10.2 Kodaira Embedding Theorem . . . . . . . . . . . . . . . . . . . . . . 112 10.3 Embedding of Riemann surfaces . . . . . . . . . . . . . . . . . . . . . 120
Selected Bibliography Including Cited Works
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123
Chapter 1 Introduction to Riemann Surfaces 1.1
Definition of Riemann Surfaces
Let X be a connected real two dimensional manifold and {Uα }α∈A an open covering of X. The coordinate charts are pairs (Uα , zα ) where zα is a homeomorphism from Uα to some Vα open subset in C. The collection of coordinate charts form an atlas of X with the compatible condition satisfied, i.e. ∀ Uα,β = Uα ∩ Uβ 6= ∅, we have a transition function fβ,α = zβ ◦zα−1 : zα (Uα,β ) → zβ (Uα,β ). Here the transition function is required to be smooth when X is a real 2-dimensional differentiable manifold. For X to be a 1-dimensional complex differentiable manifold, namely a Riemann surface, we further require transition functions be holomorphic. One shall note that two collections of coordinate charts form two atlas A1 , A2 , and they are said to be compatible if their union is still an atlas A1,2 , which is certainly larger than A1 ,A2 . keeping on enlarging the atlas by taking union of compatible atlases, we will achieve at a maximal one A0 in the sense that all the other atlases that are compatible to A0 are just part of it. And when talking about an atlas, we always refer to the maximal one and those smaller atlases contained in it are identified with each other. A maximal atlas is also called a complex structure. An interesting question is whether a manifold can have different complex structures. First of all, the answer is affirmative for C ∞ structures, e.g. Let U = R and ϕ : U → R be the identity map; let V = R and ψ : V → R be defined by ψ(t) = t3 . Then ϕ ◦ ψ −1 (t) = t1/3 is not even of class C 1 , thus (U, ϕ) and (V, ψ) are two incompatible C ∞ structures. However, the two manifolds R1 and R2 given above, with same underlying topology but different C ∞ structures are diffeomorphic, through F : R1 → R2 : F (t) = t1/3 for the local map ϕ ◦ F ◦ ψ −1 (t) = t is clearly smooth and has smooth inverse.
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Definition: A Riemann surface is a real 2-dimensional connected manifold with a complex structure on it. Locally, a Riemann surface X is just a domain in C for coordinate charts give homeomorphisms between domains in X and domains in C. ∀P ∈ X, there may be indefinitely many choices of such homeomorphisms, hence it is meaningless to talk about notions that are only for a specific coordinate chart. Instead, we are only interested in what are invariant under biholomorphic maps, e.g., the angle between two curves γ1 , γ2 on X intersecting at P , which is the same as that of z(γ1 ) and z(γ2 ) in C intersecting at z(P ). Since biholomorphic maps are conformal, this quantity is invariant under change of local coordinate z. Riemann surface is orientable. The transition function fβ,α : (xα , yα ) → (xβ , yβ ) is α 2 α 2 zα = 2i | dz | dzβ ∧d¯ zβ = | dz | dxβ ∧dyβ . orientation preserving for dxα ∧dyα = 2i dzα ∧d¯ dzβ dzβ The simplest example of Riemann surface is any domain U of C, whose complex structure is given identity map. The simplest complex Riemann surface is the extended ˆ = C ∪ {∞}, whose complex structure is given by two coordinate complex plane C ˆ − {0}, z2 = 1/z. The charts (U1 , z1 ),(U2 , z2 ) where U1 = C, z1 = z and U2 = C transition function on C − {0} is 1/z, which is clearly holomorphic. In the following, we introduce several models of Riemann surfaces.
1.2
Non-singular Algebraic Curves
If we take a Riemann surface and delete one point from it, we still get a Riemann surface albeit with holes in it. Conversely, the process can be reversed if we define a hole properly. Definition: Let X be a Riemann surface. A hole chart on X is a complex chart φ : U → V on X such that V contains an open punctured disk ∆0 := {z| 0 < |z −z0 | < } with the closure in X of φ−1 (∆0 ) inside U and this closure is transported via φ to the punctured closed disk closed disk ∆1 := {z| 0 < |z − z0 | ≤ }. In other words, a hole chart has a hole in it: the closure of ∆0 in C has z0 in it but the closure of the corresponding open set φ−1 (∆0 ) in X does not have any points corresponding to this z0 . Suppose that X is a Riemann surface with a hole chart φ : U → V on it. Let ∆0 be the open punctured disk as above and ∆ := {z| |z − z0 | < } be an open disk. ∆0 is an open subset of ∆ and is isomorphic to the open subset φ−1 (∆0 ) ⊂ X via the chart map φ. Form the identification space Z = X ∪ ∆/φ; the assumption on the closure 12
φ−1 (∆0 ) implies that Z is Hausdorff and thus is a Riemann surface, which is said to be obtained from X by plugging holes in the hole chart φ. Compactifications of certain Riemann surfaces may be effected by means of plugging holes. Suppose that X is a Riemann surface with a finite number of disjoint hole charts φi : Ui → Vi . Let Gi be the open subset φ−1 (∆0 ) in X. Suppose that X − ∪i Gi is compact. Then the surface obtained from X by plugging the holes in these hole charts is compact, since it can be decomposed as the union of finitely many compact sets, namely X − ∪i Gi and the closures of the disks which are glued in to plug the holes. ˆ The simplest example of this is the compactification of C to the Riemann sphere C. The hole chart on C is the function φ(z) = l/z, defined for z 6= 0. A more sophisticated example is the compactification of the algebraic curve given by the hyperelliptic equation y 2 = h(x), where h is a polynomial with distinct roots. Before proceeding to the compactification, we introduce the definition of non-singular algebraic curves and their closed relation with Riemann surface. Definition: An algebraic curve γ is a subset in C2 , γ = {(µ, λ) ∈ C2 |P (µ, λ) = 0}
(1.1)
Where P is an irreducible polynomial in µ and λ. In particular, it is non-singular if , ∂P ) is non-vanishing on the curve γ . the gradient ( ∂P ∂µ ∂λ Recall the implicit function theorem in complex variables: For F (µ, λ) holomorphic in µ and λ in a domain U ⊂ C2 , suppose P0 = (µ0 , λ0 ) ∈ U (µ0 , λ0 ) is non-vanishing, then in some neighborhood V ⊂ U is a root of F (., .) and ∂F ∂µ of P0 , the solution of F (µ, λ) = 0 can be written as (µ(λ), λ), i.e., F = 0 can be locally solved in λ. Moreover, locally, µ(λ) is a holomorphic function in λ, and the ∂P/∂λ derivative dµ is given by − ∂P/∂µ . dλ This enables us to construct complex structures on a non-singular algebraic curve γ in the following way: Where ∂P/∂µ 6= 0 we take λ to be the local variable and where ∂P/∂λ 6= 0 we take µ to be the local variable; the transition function µ(λ) and λ(µ) are holomorphic in their respective variables by the implicit function theorem. Hence the two charts are indeed compatible with each other and form a complex structure on γ. Therefore, non-singular algebraic curves in C2 are Riemann surfaces. Conversely, a deeper result is that every compact Riemann surface is indeed an algebraic curve, which we will show later. Hyperelliptic curves are examples of algebraic curves of importance. They are formu13
lated as µ2 =
N Y
(λ − λi ) for N ≥ 3
i=1
Clearly they are the set N Y {(µ, λ) ∈ C | P (µ, λ) := µ − (λ − λi ) = 0} 2
2
i=1
In particular, when N = 3, they are called elliptic curves. Hyperelliptic curves are non-singular if and only if λi 6= λj for any i 6= j. Indeed, when ∂P/∂µ = 0, µ must be zero and λ must QN equal to some λi , and ∂P/∂λ has a factor λ − λi only if λi is a multiple root for i=1 (λ−λi ). In the non-singular case, as a result of implicit function theorem, in the neighborhood of any (µ0 , λ0 ), where λ0 6= λi for all i = 1, 2, · · · , N , ∂P is non-vanishing and λ is taken as a local coordinate, while in the neighborhood ∂µ = 0, we have ∂P 6= 0 by the non-singular condition of any point (0, λi ) so that ∂P ∂µ ∂µ thus µ can be taken as local coordinate. Let N = 2g +2 if N is even and N = 2g +1 if N is odd, we take a transformation from µ , the hyperelliptic curve γ(µ, λ) to some curve γ 0 (m, l), by l := 1/λ and m := λg+1 Q Q 2g+2 2g+1 2 2 then the curve becomes m = l i=1 (1 − lλi ) for N odd or m = i=1 (1 − lλi ) for N even. The transformation (µ, λ) 7→ (m, l) above is a biholomorphic map from a neighborhood U∞ of {(µ, λ) ∈ γ|λ = ∞)}, where U∞ := {(µ, λ) ∈ γ||λ| > c > |λ|, ∀i} onto a punctured neighborhood V0 of (m, l) = (0, 0) (in the odd case), where V0 := {(m, l) ∈ γ 0 |0 < |l| < c−1 } or a biholomorphic map from U∞ onto two punctured neighborhoods V+ and V− of (m, l) = (±1, 0) (in the even case). Clearly, in the odd case, the local coordinate for γ 0 is µ, while in the even case, the local coordinate is λ. In conclusion, if N = 2g + 1 is odd, the curve γ has one puncture P∞ at (µ, λ) = (∞, ∞), and the local coordinate in the neighborhood of the puncture is given by the homeomorphism z∞ : (µ, λ) 7→ m(λ) =
2g+1 1 Y λi (1 − ) λ i=1 λ
!1/2
1 ≈√ λ
On the other hand, if N = 2g + 2, there are two punctures as λ → ∞, namely P+∞ µ µ with λg+1 = 1, and P−∞ with λg+1 = −1, and in neighborhoods of both punctures, the local coordinate is given by z∞ : (µ, λ) 7→ λ−1 14
Theorem: A hyperelliptic curve γ as a Riemann surface can be compactified to a compact Riemann surface γˆ by joining a point p∞ if N is odd, or by joining two points p±∞ if N is even. The local coordinates in neighborhoods of punctures are given above. Now we return to the example of compactitification of a hyperelliptic curve we have mentioned above. Assume that h has odd degree 2g + 1. Then the chart φ defined by φ(x, y) = y/xg+l is defined for |x| large, and is a hole chart on X with its hole at ∞. Plugging this hole gives a compact Riemann surface. If h has even degree 2g + 2, then we know that X has two points at infinity, denoted by ±∞. As x approaches ∞, y/xg+l approaches one of the two square roots of limx→∞ h(x)/x2g+2 . The two hole charts are φi (x, y) = 1/x for i = 1, 2, while φ1 is defined for |x| large and y/xg+1 near +∞ and φ2 is defined for |x| large and y/xg+1 near −∞. Plugging these two holes gives a compact Riemann surface. By plugging holes in Riemann surfaces, we have the following result. Theorem: All compact Riemann surfaces can be constructed from compactifications of algebraic curves.
1.3
Quotients under Group Actions
Definition: Let ∆ be a domain in C. A group G of holomorphic maps from ∆ into itself is said to be acting discontinuously on ∆ if ∀p ∈ ∆, there is a neighborhood U of p such that gU ∩ U = ∅, ∀g ∈ G and g 6= id. Two points p and p0 are said to be equivalent if for some g ∈ G we have p0 = g(p). And the quotient of this equivalence relation ∆/ ∼ is a Riemann surface. The canonical projection π : ∆ → ∆/G assigns to each point of ∆ an equivalence class [.] that it belongs to. With the quotient topology, for each point p ∈ ∆, we can have a neighborhood V such that gV ∩ V = ∅ for any g ∈ G, thus U = π(V ) is an open set in ∆/G containing [p], and π|V : V → U is a homeomorphism. The inverse of this homeomorphism z := π|−1 V is a local coordinate from the neighborhood U of [p] onto neighborhood V ⊂ C. In this way, we cover ∆/G by local coordinate charts. For any two of these coordinate charts z : U → V and z˜ : U˜ → V˜ with U ∩ U˜ 6= ∅, we have the transition function f := z˜ ◦ z −1 : V → V˜ . Since z = π|−1 ˜ = π|−1 , we have V and z V˜ −1 f = π|V˜ ◦ π|V , therefore, π|V (z) = π|V˜ (f (z)). Here z is any point of V . This shows y = f (z) ∈ V˜ and z ∈ V are in the same equivalence class, and there is some g ∈ G such that f (z) = y = g(z). Now we want to show the element g ∈ G is the same for all z ∈ V . If not suppose f (z1 ) = g1 (z1 ) and f (z2 ) = g2 (z2 ), then g1 (V ) and g2 (V ) 15
both have at least a point in V˜ , i.e. g1 (V ) ∩ g2 (V ) 6= ∅, and V ∩ g1−1 ◦ g2 (V ) 6= ∅, which contradicts the condition that G acts discontinuously on V . This shows the transition functions are indeed holomorphic and ∆/G is a Riemann surface. Now we consider the case ∆ = C and the group action G is generated by two shifts z 7→ z + w1 and z 7→ z + w2 where w1 and w2 are non-parallel complex vectors in the sense that Im w2 /w1 6= 0. G is a commutative group and consists of the elements gn,m (z) = z + n · w1 + m · w2 for n, m ∈ Z. Then the quotient C/G is a Riemann surface which is topologically a torus and is therefore compact. In fact, as a result of uniformization theorem, we have the following corollary that describes compact Riemann surfaces as quotients under group actions: Theorem: All compact Riemann surfaces can be constructed as quotients of a domain ∆ under discontinuous group actions G, namely ∆/G. Theorem(Uniformization of Compact Riemann Surfaces): Let X be a compact Rie˜ → X be the universal covering. Then mann surface of genus g and π : X ˜ =X ˜ and X is conformally equivalent to the Riemann sphere 1. If g = 0, then X ˆ := C ∪ {∞}; C ˜ is conformally equivalent to C and X is obtained by a group 2. If g = 1, then X Λ := {m · ω1 + n · ω2 : m, n ∈ Z}, with ω1 and ω2 being R-linearly independent complex numbers; ˜ is conformally equivalent to the unit disk ∆ ⊂ C and X ∼ 3. If g ≥ 2, then X = ∆/Γ, where Γ is a subgroup of Aut(∆).
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Chapter 2 Topology of Riemann Surfaces 2.1
Countable Topology
In this section, we will prove a theorem by Rad´o asserting that every Riemann surface has a countable topology (more precisely, second countable). First we introduce some lemmas about countable topology. Lemma 1: Suppose X and Y are topological spaces and f : X → Y is a continuous, open and surjective map. If X has a countable topology, so does Y . Proof: Let U be a countable basis for the topology on X and let B := {f (U )| U ∈ U} be a countable family of open subsets of Y which we claim is a basis for the topology of Y . Suppose D is an open subset of Y and y ∈ D. We want to show that there exists V ∈ B with y ∈ V ⊂ D. Since f is surjective, there exists x ∈ X with f (x) = y. The pre-image f −1 (D) is an open neighborhood of x. Hence there exists U ∈ U with x ∈ U ⊂ f −1 (D). Thus, V := f (U ) satisfies y ∈ V ⊂ D. Lemma 2(Poincar´e-Volterra): Suppose X is a connected manifold, Y is a Hausdorff space with countable topology and f : X → Y is a continuous, discrete map. Then X has a countable topology. Proof: Suppose U is a countable basis for the topology of Y . Let B be the collection of all open subsets V on X such that: (1)V has a countable topology, (2)V is the connected component of a set f −1 (U ) with U ∈ U. We claim that B is a basis for the topology on X. To see this, suppose D is an open subset of X containing a point x. We want to show that there exists V ∈ B such 17
that x ∈ V ⊂ D. Since f is discrete, there is a relatively compact open neighborhood W ⊂ D containing x so that the boundary ∂W does not meet the fibre f −1 (f (x)). Now f (∂W ) is compact and thus closed and does not contain f (x). Hence there exists a U ∈ U with f (x) ∈ U , and the intersection U ∩ f (∂W ) is non-empty. Let V be the connected component of f −1 (U ) containing x. Since V ∩ ∂W = ∅, we have V ⊂ W , therefore V has a countable topology, i.e., V ∈ B. We further claim that for every V0 ∈ B, there exists at most countably many V ∈ B having non-empty intersection with V0 . To see this, we first note that the connected components of f −1 (U ) for every U ∈ U are disjoint. Since V0 has countable topology, it can only meet countably of them. It follows from the fact that U is countable that the set {V ∈ B| V ∩ V0 6= ∅} is countable for any fixed V0 ∈ B. Now we complete the proof by showing that B is countable. We fix some V ∗ ∈ B and for every n ∈ N, we define a countable collection of sets in B, denoted by Bn ⊂ B such that Bn consists precisely of all those V∗ ∈ B with the following property: There exist a sequence of sets in B, denoted by V0 , V1 , · · · , Vn , with V0 = V ∗ , Vn = V∗ , and Vk−1 ∩ Vk 6= ∅ for k = 1, · · · , n It X is connected, it is clear that B = ∪n∈N Bn . Thus, it suffices to show that each Bn is countable, which can be done by induction on n. Indeed, B0 = {V ∗ } is clearly countable, and suppose we already know that Bk is countable for all k ≤ n. With V0 = V ∗ , we can find find at most countably many V1 with V1 ∩ V0 6= ∅, denoted by (1) (2) (i) V1 , V1 , · · · ; choosing one of them, say V1 , we can have at most countably many (i) of V2 such that V2 ∩ V1 6= ∅, etc. The process will end in n + 1 steps while in each step we can have countably many choices based on the previous one, it follows, Bn+1 is also countable, as was to be shown. This completes our induction and the whole proof of the lemma. Now we proceed to the main theorem of this section. Theorem(Rad´o): Every Riemann surface X has a countable topology. Proof: Suppose U ⊂ C is a coordinate neighborhood on X. We choose two disjoint compact disks ∆0 , ∆1 ⊂ U and set Y := X − ∆0 ∪ ∆1 . Every point of the boundary ∂Y = ∂∆0 ∪ ∂∆1 belongs to one of the two disks and Y has no intersection with ∆0 or ∆1 , it then follows that for a continuous function f : ∂Y → R with f |∂∆0 = 0 and f |∂∆1 = 1, the Dirichlet problem is solvable on Y , i.e., we can find a continuous function u : Y¯ → R, harmonic in Y , satisfying u|∂Y = f . Hence ω := d0 u is a non-trivial holomorphic 1-form on Y . Let F be a holomorphic primitive for p∗ ω on the universal covering p : Y˜ → Y . Since F is not constant, the map F : Y˜ → C is continuous and discrete, and it follows from Lemma 2 that Y˜ has a countable topology. The projection p : Y˜ → Y is continuous, open and surjective, from Lemma 1, we also have that Y has a countable topology. Since X = Y ∪ U , the topology on X is also countable, as was to be shown. 18
Definition(Partition of unity): Suppose X is a differentiable manifold and U = {Ui }i∈I is an open covering of X. Then by a differentiable partition of unity subordinate to U we mean a family (gi )i∈I of differentiable functions gi : X → R with the following properties: (1)0 ≤ gi ≤ 1 for every i ∈ I; (2)Supp(gi ) ⊂ Ui for every i ∈ I; (3)The family of supports Supp(gi ), i ∈ I, is locally finite, i.e., every point a ∈ X has a neighborhood V such that V ∩ Supp(gi ) 6= ∅ for only finitely many i ∈ I P (4) i∈I gi = 1, which is well-defined by (3). Knowing that every Riemann surface has a countable topology, we will see by the following theorem that on a Riemann surface we can always construct a partition of unity subordinate to an arbitrary open covering of the Riemann surface. Theorem: Suppose X is a differentiable manifold which has a countable topology. Then for every open covering U of X, there exists a differentiable partition of unity subordinate to U. Proof: We prove the theorem in two steps. (1)To construct a partition of unity subordinate to an open overing, we first need to show that the cover has a locally refinement. Indeed will prove a slightly stronger assertion as follows, which will simply our construction in the second step of the proof. claim: Let Br ⊂ Rn be the n-ball of radius r, and X be a countable n-dimensional manifold. Then any open covering of X admits a countable, locally finite refinement U by open subsets U that admit surjective coordinate maps ϕ : U → B2 such that the ϕ−1 (B1 ) still cover X. Proof of the claim: Since X is countable, it has at most countably many components, and we may assume X to be connected. We choose a countable covering V := {V0 , V1 , · · · } of X by open sets with compact closures. Define by induction a sequence of compact sets A0 ⊂ A1 ⊂ A2 ⊂ · · · as follows: Set A0 := V0 , and suppose all the Ai for i ≤ k has been defined. Denoted by Ji the smallest index such that Ai ⊂ ∪j≤Ji Vj for i = 0, 1, 2, · · · , k We set Ak+1 := ∪j≤Jk Vj . If there exists some n0 ∈ N such that Jn = Jn+1 = · · · , for all n ≤ n0 , then An0 is closed (and compact)and open in X, hence An0 = X for X is 19
connected. Otherwise, as Jk → ∞, we also have ∪i Ai = ∪i Vi = X In both cases, we have X = ∪i Ai . ˚i+2 − Ai to Let W be an open covering of X. Intersect all open set W ∈ W with all A 0 construct a new open covering W that refines W. For each x ∈ X, find a coordinate neighborhood Ux contained in some element of W 0 together with a surjective local coordinate ϕ : Ux → B2 . Define Ux0 := ϕ−1 (B1 ). ˚i−1 by finitely many of the For each i, we choose a covering of the compact set Ai − A 0 Ux and denote by Ki the finite set that consists precisely of the x we picked up. Then K := ∪i Ki is countable. Consider the covering U by all Ux , x ∈ K. This covering is a refinement of W, and is locally finite. To see this, we note that the open sets ˚i+1 − Ai−1 form an open covering of X and each can intersect only finitely many A Ux , for x ∈
i+2 [
Kj
j=i−2
Clear our open sets are coordinate neighborhoods, as desired. (2)We find a locally finite covering U := {Ux }x∈K as we did in the first step, together with coordinate maps ϕx with respect to each x, which can be chose C ∞ if X is a differentiable manifold. Let h ≥ 0 be a C ∞ function on Rn with support in B2 and is strictly positive on B1 . Define hx for x ∈ K by hx := h ◦ ϕx . Since hx has compact support, it is the restriction of a continuous function on X, being C ∞ if X P is differentiable, with support in Ux (still denoted by hx ). Now the function g := x∈K hx is a strictly positive function on X; the sum exists and is continuous (C ∞ if X is so) because the covering U is locally finite. The functions gx , x ∈ K, defined by gx := hx /g form the desired continuous partition of unity, which is C ∞ if X is differentiable. From now on, we will make use of the results in this section in some proofs when a partition of unity is needed.
2.2
Compact Riemann Surface as Sphere with Handles
A Riemann surface is a real 2-dimensional orientable smooth manifold. Recall that for every Riemann sphere, we can apply the compactification method to get a compact Riemann surface. In this section we focus on the topology of the compact Riemann 20
surface, and start with a classification scheme by introducing the model of sphere with handles. At the end of the section, we are able to present the Riemann-Hurwitz Theorem. Theorem: Any compact Riemann surface X is homeomorphic to a sphere with handles. The number g ∈ N of handles is called the genus of X. Conversely, two manifolds with different genera are not homeomorphic.
In fact, in the case where the Riemann surface is a hyperelliptic curve, after compactification, the genus of it is g, for N = 2g + 1 being odd or N = 2g + 2 being even. Now we introduce a planar image for these concepts.
Proposition: Let Πg be an extended plane(namely, R2 ∪ {∞} which is homeomorphic to S 2 ) with 2g holes bounded by 2g non-intersecting curves γ1 , γ10 , · · · , γg , γg0 . Here for each pair γi , γi0 , we topologically identify them while their orientations with respect to Πg are opposite. Then Πg is homeomorphic to sphere with g handles. Intuitively, one can imagine there is a sphere with g handles and to obtain an object in the extended plane or Riemann sphere, it suffices to cut up all the handles thus there is a sphere with 2g holes, grouped into g pairs. For each pair, the two boundaries can be glued together.
Usually, a sphere with g handles can be described, up to homeomorphism in the following way: let ∆ be a 4g-gon with 4g consecutive edges a1 , b1 , a01 , b01 , · · · , ag , bg , a0g , b0g in the order they traverse the boundary of ∆. ai , a0i and bi , b0i are identified pairs of opposite orientations with respect to ∆. Under this identification, we have a glued surface X, with all the vertices of ∆ mapping onto a single point p ∈ X. Meanwhile ai , bi map onto two closed curves through p and a0i , b0i map to the same curves as ai , bi with reversed direction. Then ∆ is homeomorphic to a sphere with g handles. In particular, a sphere without handles, namely g = 0, is homeomorphic to 2-gon with edges a, a0 .
It is important that a compact Riemann surface X is finite triangulable. This enables us to talk about Euler characteristics χ = F − E + V of a compact Riemann surface, which is invariant under different triangulations. In fact χ is a topological invariant, which is Rnot obvious at all. It can be proved with the help of Gauss-Bonnet Theorem 1 K (No boundary term due to the compactness of X). Here K is the χ = 2π X Gaussian curvature of X, which is an intrinsic geometric quantity. From the right hand side, we see that χ is indeed independent of the triangulation.
21
2.3
Fundamental Groups
A curve in a topological space X is a continuous map γ : I → X where I := [0, 1] ⊂ R. Two points on p = γ(0) and q = γ(1) on X are respectively the initial and the end point. For specific points p,q on X, there are different choices of curves that connect them. Two such curves γ0 , γ1 are called homotopic if one can be deformed continuously to the other, through the following continuous map A : I × I → X such that A(t, 0) = γ0 (t) and A(t, 1) = γ1 (t) for every t ∈ I, and A(0, s) = p and A(1, s) = q for every s ∈ I. If we fix a certain s ∈ I, then A(t, s) = γs (t) is again a curve from p to q. And the family of curves {γs }0≤s≤1 is called a deformation or a homotopic from γ0 to γ1 . In fact, the notion of homotopy defines an equivalence relation: the reflexivity is clear; the symmetry holds for a deformation A : I × I → X from γ0 to γ1 induces a deformation A0 from γ1 to γ0 by reversing the second variable in A, i.e. A0 = (id, φ)◦A, where φ(s) = 1 − s, ∀ s ∈ I. To show the transitivity, suppose A : I × I → X and B : I × I → Xare respectively the deformations from γ0 to γ1 and that from γ1 to γ2 . By assumption, we have A(t, 0) = γ0 (t), A(t, 1) = B(t, 0) = γ1 (t), B(t, 1) = γ2 (t), and A(0, s) = B(0, s) = p, A(1, s) = B(1, s) = q. Thus we can define a deformation from γ0 to γ2 , C : I × I → X with C(t, s) = A(t, 2s) for 0 ≤ s ≤ 1/2 and C(t, s) = A(t, 2s − 1) for 1/2 ≤ s ≤ 1. Hence, γ0 ∼ γ1 , γ1 ∼ γ2 ⇒ γ0 ∼ γ2 . Therefore a curve connecting p, q always refers to an equivalence class of curves from p to q under the homotopic equivalence. Let γ1 γ2 be curves on X with γ1 (1) = γ2 (0), the product of them is defined as (γ1 · γ2 )(t) = γ1 (2t) for 0 ≤ t ≤ 1/2 and γ2 (2t − 1) for 1/2 ≤ t ≤ 1. The inverse of a curve γ(t) is defined as γ(t)1 −1 = γ(1 − t). These operations apply also to the equivalence class the curve represents. A curve with same initial and end points is called a closed curve. A closed curve with initial (end) point a ∈ X is said to be null-homotopic if it is homotopic to a constant curve at a, i.e. the curve can be continuously deformed to the single point. Any two closed curves through a point a can be multiplied and the set π1 (X, a) of homotopy classes of closed curves in X with initial and end point a, forms a group under the operations define above, which is named fundamental group of X with base point a. The identity element in this group is the class of curves that can be continuously contracted to the base point. For two points a,b on X, and the fundamental groups π1 (X, a), π1 (X, b), if there is a curve w in X connecting a, b, there is an isomorphism f from π1 (X, a) to π1 (X, b), ∀{γa } ∈ π1 (X, a) where γa is a representation of a homotopy class, we have {w−1 · γa · w} ∈ π1 (X, b). It is easy to see that this map is indeed a canonical isomorphism between π1 (X, a) and π1 (X, b). In an arcwise connected topological space X, the fundamental group is independent of the choice of base point, hence is simply denoted by π(X). It is clear, if further 22
the space is simply connected, there is only the identity element in the fundamental group. In this case all curves connecting two distinct points of X are also in one single homotopy class. Note that a Riemann surface is locally homeomorphic to C ≡ R2 , which is locally arcwise connected; moreover, a connected, locally arcwise connected topogical space is also globally arcwise connected, hence we do not specify the base point for the fundamental group of a Riemann surface. Now we return to the compact Riemann surface X of genus g and the model of gluing the 4g-gon ∆. a1 , b1 , a01 , b01 , · · · , ag , bg , a0g , b0g are closed curves on X with same base point. In particular, a0i = a−1 and b0i = b−1 i i , where the inverse is defined for curves. Q g −1 −1 is theQoriented boundary of ∆, which maps to the base point in i=1 ai · bi · ai · bi −1 the gluing scheme. Hence gi=1 Ai · Bi · A−1 = 1, where Ai ,Bi are the homotopy i · Bi classes for ai ,bi , 1 is the identity in π1 (X), and the inverse here is the for homotopy classes. In fact, these curves form a basis of first fundamental group of X over Z, which we will discuss in the next section.
2.4
First Homology Groups
To start with, we introduce the chain complex on Riemann surfaces. Definition: A chain complex C = {C1 , ∂q } is a sequence of abelian groups Cq (qdimensional chain group), and a sequence of group homomorphisms ∂q : Cq → Cq−1 (q-dimensional boundary operator), ∂q+1
∂q
∂q−1
· · · → Cq+1 → Cq → Cq−1 → Cq−2 → · · ·
(2.1)
with ∂q ◦ ∂q+1 = ∂ 2 = 0 for every q. P Let X ni · γi P be a Riemann surface with triangulation. The formal sums Σni · pi , and ni · Di are called 0-chain, 1-chain, 2-chain, where ni ∈ Z,pi is any point on X, γi is any curve on X and Di is any oriented triangle in the triangulation. The set of i-chains is called i-th chain class, denoted by Ci (X) respectively. Each of these chain classes form an abelian group in a natural way (with the formal addition). In fact, any 0-chain is a divisor on X, which we will discuss in detail in the next chapter. The boundary operator ∂ : Ci+1 → Ci is defined in the following way: Let γp,q be a curve on X with initial point p and end point q, then ∂1 γp,q = qP −p ∈ C0 , and P the operation on a 1-chain follows this definition by linearity, namely, ∂1 ni ·γi = ni ·(∂γi ). Similarly, for an oriented triangle D0 with vertices (p1 , p2 , p3 ) and oriented edges γp1 ,p2 , γp2 ,p3 , γp3 ,p1 , then we have ∂2 D0 = γp1 ,p2 + γp2 ,p3 + γp3 ,p1 ∈ C1 . And the operation extends to C2 by linearity. 23
In the study of first homology, we focus on 1-chains. In C1 , we define the subgroup ker∂1 of C1 to be 1-cycles of C1 , denoted by Z1 (X), which contain 1-chains whose end point of each curve cancels with the initial point of another curve. Meanwhile, we define the subgroup Im{∂2 : C2 → C1 } to be the 1-boundaries, denoted by B1 . As term from homology theory, cycles are also called closed chains. Clearly, B1 (X) ⊂ Z1 (X) ⊂ C1 (X), as a result of ∂ 2 = 0: ∂1 (∂2 (p1 , p2 , p3 )) = ∂1 (γp1 ,p2 + γp2 ,p3 + γp3 ,p1 ) = p2 − p1 + p3 − p2 + p1 − p3 = 0. The quotient group H1 (X) := Z1 (X)/B1 (X) is defined to be the first homology group of the Riemann surface X. Two 1-chains γ1 , γ2 are said to be homologous if γ1 − γ2 ∈ B1 (X), and [γ1 ] = [γ2 ], where [γ] ∈ H1 (X) is the equivalence class that γ represents. More precisely, we use the notation H1 (X, Z) since the coefficients are in Z. The homology groups thus defined are actually independent of the triangulation. It can be shown, any From above, we shall see that first homology group H1 (X, Z) and fundamental group π1 (X) of a Riemann surface X are closely related (here we omit the base point since Riemann surface is arcwise connected). More generally, when X is a connected simplicial complex, they both describe the number of holes on X. However, H1 (X, Z) is an abelian group while π1 (X) is not abelian in general. The following theorem precisely clarify their connection. Theorem: For a connected simplicial complex X, the first homology group H1 (X) is π1 (X) . isomorphic to the abelianization of the fundamental group of X, namely [π1 (X),π 1 (X)] The proof of this theorem is non-trivial and we will need some properties of simplicial complex.
24
Chapter 3 Sheaves and Cohomology 3.1
Sheaves
Definition: let X be a topological space, a presheaf of abelian groups on X is an assignment to each each open set U ⊂ X an abelian group F(U ) (By default F(∅) = {0}) with group homomorphisms ρUV : F(U ) → F(V ) for any open sets V ⊆ U with the following properties: (1)ρUU = idF (U ) , ∀U ⊂ X; (2)For open set W ⊂ V ⊂ U , we have ρUW = ρVW ◦ ρUV Definition: A presheaf is called a sheaf for every open set U ⊂ X and an arbitrary open covering {Ui }i∈I of U , the followings are satisfied: (1)If f ,g ∈ F(U ) and f |Ui = g|Ui for any i, then we have f ≡ g; (2)For each i, suppose we have an element fi ∈ F(Ui ) with fi |Ui ∩ Uj = fj |Ui ∩ Uj , ∀i, j ∈ I, we can find some f ∈ F(U ) such that f |Ui = fi . Condition (2) is about the existence of a global sheaf on U with certain local data while condition (1) guarantees the uniqueness of such a sheaf. S In order to construct a sheaf from presheaf F(U ), we take the disjoint union a∈U F(U ) and introduce an equivalence relation by identifying any f ∈ F(U ) and g ∈ F(V ) if and only if a neighborhood of ∃W ⊂ U ∩ V such that f |W = g|W (intuitively, we are identifying elements with local behaviors at a point, or in terms of direct limit, they will finally become equal in the direct system). The set of equivalence classes, denoted by Fa is called stalk of presheaf F at a. Then the stalk Fa with the operation defined on the equivalence classes by means of the operation defined on the representatives also form an abelian group. For an element f in F(U ), the map ρa assigns to it an equivalence class it belongs to in Fa . The map is called the germ of f at point a. 25
Then we can define a topology on |F| =
S
a∈X
Fa in the following way:
Let U be an arbitrary open set and we define f ∈ F(U ), [U, f ] := {ρx (f ) : x ∈ U }. The collection of [U, f ] forms a basis for the topology on |F|. And the projection pr : |F| → X, which assigns to each stalk Fx the point x, is a local homeomorphism. To see this, suppose ϕ ∈ |F| and pr(ϕ) = x, there exists an [U, f ] which contains ϕ; It is an open neighborhood for ϕ, while U is an open neighborhood for x. The projection map from restricted to [U, f ] has its image as U , and clearly this map is injective, hence is bijective; An open set V ⊂ U has its preimage [V, f ] is also in the basis define above, hence is open; Conversely, any open set in [U, f ] is union of S elements in the basis {[Uα , fα ]α∈A }, with Uα ⊂ U , hence the image of the open set is α∈A Uα , which is clearly an open set in U . Hence this map is both open and continuous, thus it is a homeomorphism. The topological space thus defined for |F| is a family of abelian groups. For any openSneighborhood U , |F|(U ) is the collection of continuous maps s : U → |F| : s(U ) = i∈I [Ui , fi ] where {Ui }i∈I is an open covering of U , and pr ◦ s is the identity map from |F| to itself. And the homomorphism ρ : U → V defined for this sheaf is the restriction of s : U → |F| onto subset V . The morphism between presheaves is defined in the following way: Suppose we have two presheaves F, G on X, V ⊂ U ⊂ X are open subsets. The group homomorphisms are ρUV and rVU respectively. Let α be the morphism from F to G, then the following diagram commutes: αU F(U ) −−− → G(U ) ρU rU (3.1) yV yV α
V F(U ) −−− → G(V )
If we define kernel of the morphism to be the presheaf {ker(αU ), ρUV |ker(αU )}, it is clear that if both F and G are sheaves, the kernel so defined is also a sheaf. Consider U1 ,U2 , with V = U1 ∩ U2 , the zeros in group G(U1 ) and G(U1 ) are both mapped to the zero in G(U1 ∩ U2 ) through rUU11∩U2 and rUU12∩U2 respectively. Since the diagram commutes, it follows given f1 ∈ αU1 and f2 ∈ αU2 , ρUU11 ∩U2 (f1 ) and ρUU11 ∩U2 (f2 ) are both in ker(αU1 ∩U2 ). If ρUU11 ∩U2 (f1 ) = ρUU11 ∩U2 (f2 ), there exists some F1 ∈ F(U1 ∪ U2 ) such that, ρUU11 ∪U2 (F1 ) = f1 ; Similarly, there exists some F2 ∈ F(U1 ∪ U2 ) 2 such that ρUU12 ∪U2 (F1 ) = f2 . By the property of presheaf, we have ρUU11 ∪U ∩U2 (F1 ) = 2 ρUU11 ∩U2 ◦ ρUU11 ∪U2 (F1 ) = ρUU21 ∩U2 ◦ ρUU12 ∪U2 (F2 ) = ρUU11 ∪U ∩U2 (F2 ), i.e. F1 = F2 . Such an element in F(U1 ∪U2 ) does exist for F is a sheaf. It remains to show this F = F1 = F2 is indeed an element in the abelian group ker(αU1 ∪U2 ). This will make use of the condition that G is a sheaf. αU1 ∪U2 (F ) = G ∈ G(U1 ∪ U2 ), and since the diagram commutes, we have rUU11 ∪U2 (G) = rUU21 ∪U2 (G) = 0, which implies G = 0 and F ∈ αU1 ∪U2 . This shows the 2nd condition of sheaf is satisfied, while the 1st condition (uniqueness) follows from the fact F1 = F2 . Hence the kernel of α is indeed a sheaf. We define the image of the morphism to be the presheaf {Im(αU , rVU |Im(αU )}, this is not necessarily a sheaf, even if both F and G are sheaves. However, associate to 26
every presheaf, we can construct a sheaf by disjoint union and taking the direct limit, hence by image of α between two sheaves, we always mean the corresponding sheaf.
3.2
Cohomology on Sheaves
Now we introduce the definition of cohomology on sheaves. let X beSa topological space (e.g. Riemann surface), F a sheaf of abelian groups on X, U = i∈I Ui be an open covering of X. For q = 0, 1, 2, · · · , the group of q-cochains of F relative to U is defined as: Y C q (U, F) := F(Ui0 ∩ · · · ∩ Ui1 ) (3.2) (i0 ,··· ,iq )∈I q+1
This means, an element in the q-cochain group is a family (fi0 ,··· ,iq )i0 ,··· ,iq , where each fi0 ,··· ,iq ∈ F(Ui0 ∩ · · · ∩ Uiq ) and the index takes over all (i0 , · · · , iq ) ∈ I q+1 . Suppose |I q+1 | = C, where the cardinality of the index set C may not be finite, the addition of two q-cochains act componentwisely as addition between two elements in some F(Ui0 ∩ · · · ∩ Uiq ). We introduce the coboundary operator δ : C 0 (U, F) → C 1 (U, F) → C 2 (U, F) For any 0-cochain (fi )i∈I ∈ C 0 (U, F), it is mapped to (gij )i,j∈I ∈ C 1 (U, F), with gij = fj |Ui ∩Uj − fi |Ui ∩Uj Similarly, for any (fij )i,j∈I ∈ C 1 (U, F), it is mapped to (gijk )i,j,k∈I ∈ C 2 (U, F), with gijk = fjk |Ui ∩Uj ∩Uk − fki |Ui ∩Uj ∩Uk + fij |Ui ∩Uj ∩Uk Clearly, coboundary operators are homomorphisms between cochain groups. In particular, we define two subgroups of C 1 (U, F). The group of 1-cocycles, Z 1 (U, F) := Ker{C 1 (U, F) → C 2 (U, F)} The group of 1-coboundaries, B 1 (U, F) := Image{C 0 (U, F) → C 1 (U, F)} We note that an 1-cocycle is precisely an 1-cochain (fij ) ∈ C 1 (U, F) satisfying the following cocycle relation fik = fij + fjk on Ui ∩ Uj ∩ Uk 27
(3.3)
for all i, j, k ∈ I. We obtain from the relation (3.3) fii = 0, fij = −fji by setting i = j = k. Clearly, an 1-coboundary in B 1 (U, F) lies in Z 1 ((U ), F), which is called a splitting cocycle. In other words, an 1-cocycle (fij ) ∈ Z 1 (U, F) splits precisely if there is a 0-cochain (gi ) ∈ C 0 (U, F) such that fij = gi − gj on Ui ∩ Uj
(3.4)
for every i, j ∈ I. An 1-coboundary group is a normal sub-group of 1-cocycle group, and by taking the quotient, we define the first cohomology group of F with respect to U: Z 1 (U, F) H (U, F) = 1 B (U, F) 1
(3.5)
Clearly, it is still an abelian group. An element of H 1 (U, F) is a class of 1-cocycles that mutually differ by an 1-coboundary. Two cocycles belonging to a same cohomology class are said to be cohomologous. We note that this definition depends on the covering U and our goal is to define cohomology groups H 1 (X, F) independent of the covering, but only related to the sheaf F of abelian groups on X. Intuitively, to obtain such a covering-independent definition, we have to refine the covering and take a limit. An open covering B = {Vk }k∈K is called finer than the covering U = {Ui }i∈I , denoted B < U, if every Vk ⊂ Ui for some i ∈ I. Under this circumstance, there is a mapping between two index sets, τ : K → I such that Vk ⊂ Uτ (k) for every k ∈ K. It induces the following map between cocycles τ ∗ : Z 1 (U, F) → Z 1 (V, F) by mapping any (fij ) ∈ Z 1 (U, F) onto some (gkl ) ∈ Z 1 (V, F) such that gkl = fτ (k),τ (l) |Vk ∩Vl for every k, l ∈ K. Since the mapping takes coboundaries onto coboundaries, it induces a natural homomorphism of the corresponding cohomology group, which will also be denoted by τ ∗ . Lemma 1: Let τ, σ : K → I be two refinement maps, the induced map τ ∗ , σ ∗ : H 1 (U, F) → H 1 (V, F) 28
is independent of the choice of the refinement. Proof: For every k ∈ K, we have Vk ⊂ Uτ (k) ∩ Uσ(k) , hence we can define hk = 0 fτ (k),σ(k) |Vk ∈ F(Vk ). Suppose (fij ) ∈ Z 1 (U, F) and let gkl = fτ (k),τ (l) |Vk ∩Vl , gkl = fσ(k),σ(l) |Vk ∩Vl . Then we have, on Vk ∩ Vl , 0 gkl − gkl = fτ (k),τ (l) − fτ (k),σ(l) + fτ (k),σ(l) − fσ(k),σ(l)
= fσ(l),τ (l) − fσ(k),τ (k) = hk − hl ∈ B 1 (V, F) 0 Thus (gkl ) and (gkl ) are cohomologous as desired.
Lemma 2: Let B be a refinement of U. The induced homomorphism H 1 (U, F) → H 1 (V, F) is injective. Proof: Let τ : K → I be a refinement. Let ξ = {(fij )i,j∈I } ∈ Z 1 (U, F) be a cocycle such that τ ∗ (ξ) belongs to the cohomology class [0] in B 1 (V, F). Thus there exists gk ∈ F(Vk ) for every k ∈ K such that fτ (k),τ (l) = gk − gl on Vk ∩ Vl . Therefore, on Ui ∩ Vk ∩ Vl , we have gk − gl = fτ (k),τ (l) = fτ (k),i + fi,τ (l) which can be re-written as gk + fi,τ (k) = gl + fi,τ (l) Then for a fixed i, we have a family of functions on sets (U1 ∩ Vk )k∈K , which, by sheaf axiom, can be patched to form an hi ∈ F(Ui ), such that hi = fi,τ (k) + gk on Ui ∩ Vk and thus, on Ui ∩ Uj ∩ Vk , we have fij = fi,τ (k) + fτ (k),j = fi,τ k + gk − gk − gj,τ (k) = hi − hj Since k is arbitrary, it follows, by sheaf axiom again that fij splits on Ui ∩ Uj , so is the cocycle ξ = (fij ) with respect to U, as was to be shown. Now we define the first cohomology group H 1 (X, F) with respect to a sheaf of abelian groups on X. Let U = {Ui }i∈I ,V = {Vk }k∈K be two coverings of X, V < U as before. There is a homomorphism τ (U, V) : H 1 (U, F) → H 1 (V, F) induced by a refinement τ but independent of our choice. If W is a refinement of V, we have τ (U, W) = τ (V, W) ◦ τ (U, V). Thus we can define the`following equivalence relation on the disjoint union of the H 1 (U, F), denoted by U H 1 (U, F), with U 29
running through all open coverings of X. Two cohomology classes ξ ∈ H 1 (U1 , F) and η ∈ H 1 (U2 , F) are said to be equivalent, if there exists an open covering B, which is a refinement to both U1 (by τ1 ) and U2 (by τ2 ) such that τ1 (U1 , B)(ξ) = τ2 (U2 , B)(η) The set of equivalence classes is called the direct limit of the system {H 1 (U, F)}, and is called the first cohomology group of X with coefficients in the sheaf F,i.e., ! a H 1 (X, F) := lim H 1 (U, F) = H 1 (U, F) / ∼ (3.6) U
U
For any covering U, there is a map τU : H 1 ((U ), F) → H 1 (X, F), taking any ξ ∈ H 1 ((U ), F) onto its equivalence class. And from Lemma 2, we know that τU is injective. And in particular, H 1 (X, F) = 0 if H 1 (U, F) = 0 for evering open covering of X. To define addition in H 1 (X, F), consider two equivalence classes x, y ∈ H 1 (X, F), represented by ξ ∈ H 1 (U1 , F) and η ∈ H 1 (U2 , F) respectively. Let B be a common refinement to both U1 and U2 , then we define, x + y := τ (U1 , B)(ξ) + τ (U2 , B)(η) which is independent of the choice of refinements. Theorem(Leray):Let F be a sheaf of abelian groups on the topological space X. Let U = {Ui }i∈I be an open covering of X. Suppose H 1 (Ui , F) = 0 for all i ∈ I. Then the natural map H 1 (U, F) → H 1 (X, F) is an isomorphism. Proof: As we have seen, by Lemma 2, this map is injective, and we are left to show the surjectivity of it. For this purpose, we consider an arbitrary refinement V = {Vα }α∈A of U by τ := τ (U, V) : A → I. It suffices to prove that the induced map τ ∗ : H 1 (U, F) → H 1 (V, F) is surjective. In other words, given a cocycle (fαβ ) ∈ Z 1 (V, F), we want to find a cocycle (Fij ) ∈ Z 1 (U, F) such that the cocycles (Fτ (α),τ (β) ) and (fαβ ) are cohomologous relative to V, and in this sense, an induced map τ ∗ whose image contains (fαβ ) is well-defined, from which we conclude that the map is surjective. We note that Ui ∩ V := {Ui ∩ Vα }α∈A is an open covering of Ui and by assumption, H 1 (Ui , F) = 0, i.e. (fαβ )|Ui splits under the covering Ui ∩ V. Thus, for any α, β ∈ A, we have fαβ = giα − giβ on Ui ∩ Vα ∩ Vβ for some giα ∈ F(Ui ∩ Vα ) for every α ∈ A. Now, on the intersection Ui ∩ Uj ∩ Vα ∩ Vβ for all α, β ∈ A, we have fαβ |Ui ∩Uj = gjα − gjβ = giα − giβ 30
Thus, by sheaf axiom, there exists some Fij ∈ F(Ui ∩ Uj ) such that fij = gjα − giβ on Ui ∩ Uj ∩ Vα Clearly, the family of such Fij , denoted by (Fij ) := {(Fij )}i,j∈I satisfies the cocycle relation, i.e., (Fij ) ∈ Z 1 (U, F). Indeed, on Ui ∩ Uj ∩ Uk ∩ Vα for every α ∈ A, we have Fij + Fjk + Fki = gjα − giα + gkα − gjα + giα − gkα = 0 Applying sheaf axiom, we see that (Fij ) splits with respect to U. Now on Vαβ we have Fτ (α),τ (β) − fαβ = (gτ (β),α − g − τ (α), α) = gτ (β),β − gτ (α),α where, we note gτ (α),α ∈ F(Uτ (α) ∩ Vα ) = F(Vα ). Thus we see that (Fτ (α),τ (β − fα,β ) splits relative to V, i.e., (Fτ (α),τ (β) ) and (fαβ ) are cohomologous. To end this section, we make some remarks about the zeroth cohomology group. Suppose F is a sheaf of abelian groups on the topological space X and U := {Ui }i∈I is an open covering of X. We define, The group of 0-cocycles, Z 0 (U, F) := Ker{C 0 (U, F) → C 1 (U, F)} The group of 0-coboundaries, B 0 (U, F) := 0 Zeroth cohomology group, H 1 (U, F) := z 0 (U, F)/B 0 (U, F) = Z 0 (U, F) From the definition of δ it follows that the a 0-cochain (fi ) ∈ C 0 (U, F) belongs to the 0-cocycle precisely if fi |Ui ∩Uj = fj |Ui ∩Uj for every i, j ∈ I. By sheaf axiom, the elements of (fi ) piece together to form a global element f ∈ F(X) and there is a natural isomorphism H 0 (U, F) = Z 0 (U, F) ∼ = F(X) Therefore, all the groups H 0 (U, F) are isomorphic and are independent of the covering U, which enables us to define the zeroth cohomology group as H 0 (X, F) := F(X)
3.3
Examples and Applications
Theorem: Suppose X is a Riemann surface and E is the sheaf of differentiable functions on X. Then H 1 (X, E) = 0. Proof: Suppose U := {Ui }i∈I is an arbitrary open covering of X. To show that 31
H 1 (X, E) = 0, it suffices to show that H 1 (U, E) = 0 for any open covering U. As we have seen from Chapter 2, there is a partition of unity subordinate to U, i.e., a family (ψi )i∈I of differentiable functions on X with following property: (1)Supp(ψi ) ⊂ Ui for every i ∈ I; (2)Every point of X has a neighborhood meeting only finitely many of the sets Supp(ψ P i ); (3) i∈I ψi = 1. To show H 1 (U, E) = 0, let (fij ) ∈ Z 1 (U, E), and we want to show that (fij ) splits relative to U. For this purpose, consider the function ψj · fij , which is defined on Ui ∩ Uj . By assigning it the value zero outside Supp(ψi ), we can extend this function to all if Ui , which will still be denoted by ψj · fij . It now belongs to E(Ui ). We set gi :=
X
ψj · fij
i∈I
on Ui , where, by property (2) of partition of unity, the sum is taken over finitely many non-zero terms in a neighborhood of any point of Ui , thus this function is well-defined. It then follows gi ∈ E. For i, j ∈ I, we have X X X X ψk · fij = fij ψk · (fik − fjk ) = ψk · fjk = ψk · fik − gi − gj = k∈I
k∈I
k
k
where the third identity follows from the cocycle relation and the last follows from the property (4) of partition of unity. Since fij splits on Ui ∩ Uj for every i, j ∈ I, (fij ) ∈ B 1 (U, D). Therefore, H 1 (U, E) = 0, as was to be shown. Similarly, we also have H 1 (X, E (1) ) = H 1 (X, E 1,0 ) = H 1 (X, E 0,1 ) = H 1 (X, E (2) ) = 0. Theorem: Suppose X is a simply connected Riemann surface, then we have H 1 (X, C) = H 1 (X, Z) = 0, where, C (resp. Z) denotes the sheaf of locally constant C-valued (reps. Z-valued) functions. Proof: For the first part, suppose U := {U }i∈I is an open covering of X, and (cij ) ∈ Z 1 (U, C). Since Z 1 (U, C) ⊂ Z 1 (U, E) and H 1 (U, E) = 0, there exists a 1cochain (fi ) ∈ C 0 (U, E), such that cij = fi − fj on Ui ∩ Uj Since cij is a constant function on Ui ∩ Uj , dcij = 0, i.e., dfi = dfj . It follows from sheaf axiom that there exists a global differential form ω ∈ E (1) with ω|Ui = dfi , which is closed since d(ω|Ui ) = d2 fi = 0 for every i ∈ I. Since X is simply connected, there is a primitive f ∈ E such that df = ω. Now we set ci := fi − f |Ui , it follows dci = dfi − df = 0 on Ui . Therefore ci is locally constant, i.e. (ci ) ∈ C 0 (U, C). Moreover, on Ui ∩ Uj , we have cij = fi − fj = (fi − f ) − (fj − f ) = ci − cj 32
It follows that (cij ) splits relative to U in the sheaf C. To prove the second part of the theorem, suppose (aij ) ∈ Z 1 (U, Z). Since Z 1 (U, Z) ⊂ Z 1 (U, C), and H 1 (U, C), there exists a cochain (ci ) ∈ C 0 (U, C) such that aij = ci − cj on Ui ∩ Uj On the other hand, since aij is integer-valued, exp(2πi·aij ) = 1, whence exp(2πi·ci ) = exp(2πi · cj ) on Ui ∩ Uj . Thus, there is a global constant function b ∈ C∗ with b = exp(2πi · ci ) for every i ∈ I. We choose some c ∈ C such that b = exp(2πi · c) and set ai := ci − c on each Ui . Since exp(2πi · ai ) = exp(2πi · ci ) · exp(−2πi · c) = 1, each ai is integer-valued, i.e., (ai ) ∈ C 0 (U, Z). Moreover, aij = ci − cj = (ci − c) − (cj − c) = ai − aj It follows that (aij ) splits relative to U in the sheaf Z. As an application of Leray’s theorem, by setting X := C∗ , we have the following result H 1 (C∗ , Z) = Z Proof: Let U1 := C∗ − R+ and U2 := C∗ − R− . U := {U1 , U2 } is an open covering of C∗ . Since Ui are simply connected, H 1 (Ui , Z) = 0 for i = 1, 2 as we have shown. By Leray’s theorem, we have H 1 (U, Z) ∼ = H 1 (X, Z) := H 1 (C∗ , Z) For any cocycle (aij ) ∈ Z 1 (U, Z), it follows from the cocycle relation that aii = 0 and aij = −aji , thus Z 1 (U, Z) is completely determined by the element a12 ∈ Z(U1 ∩ U2 ), i.e., Z 1 (U, Z) and Z(U1 ∩ U2 ) are isomorphic. Since U1 ∩ U2 consists of two connected components, namely the two half planes, we are free to assign two integer values on each of them for any locally constant integer-valued functions. It then follows that Z(U1 ∩ U2 ) ∼ = Z × Z. Since Ui is connected, we also have Z(Ui ) ∼ = Z, hence 0 ∼ C (U, Z) = Z × Z. On the other hand, since U only contains two open sets, the second order boundary map takes all 1-cochains to zero, i.e. Z 1 (U, Z) := Ker{δ : C 1 (U, F) → C 2 (U, F)} = C 1 (U, Z) so the boundary map δ :Z×Z∼ = C 0 (U, F) → C 1 (U, Z) ≡ Z 1 (U, Z) ∼ =Z×Z is given by (b1 , b2 ) 7→ (b1 − b2 , b1 − b2 ). Thus B 1 (U, Z) consists exactly of those (a1 , a2 ) ∈ Z × Z with a1 = a2 . Hence H 1 (U, Z) ∼ = Z × Z/Z ≡ Z. Similarly, we have H 1 (C∗ , C) ∼ = C based on the fact that H 1 (Ui , C) = 0 for i = 1, 2 and by Leray’s theorem. 33
In the following part, we introduce a theorem by Mittage-Leffler, which plays an important role, connecting Leray’s theorem with the theory of Riemann surfaces. Theorem(Mittag-Leffler): Let U be an open set in C, then H 1 (U, O), where O is the sheaf of germs of holomorphic functions on U . To prove the theorem, we first introduce a lemma about the solvability of ∂¯ := equations in an open set on the complex plane.
∂ ∂ z¯
Lemma: Let U be an open set in C and let f ∈ C ∞ (U ). There exists u ∈ C ∞ (U ) ¯ = f. such that ∂u Proof of the lemma: Recall, writing z = x + iy with x, y ∈ R, we have 1 ∂ ∂ ∂ 1 ∂ ∂ ∂ = +i , = −i ∂z 2 ∂x ∂y ∂ z¯ 2 ∂x ∂y Assume that f has a compact support in U , and define Z 1 f (z + w) u(z) = dw ∧ dw¯ 2πi C w ¯ = f . To see this Then u is a smooth function on C satisfying the condition that ∂u 1 we first note that |w| is integrable on any compact set ∆ ∈ C since Z Z 1 dw ∧ dw¯ = dr ∧ d cos θ ∆ ∆ |w| (z) in polar coordinates at the origin. Since f has compact support, limh∈R,h→0 f (z+h)−f = h ∂f (z) is uniformly bounded in Supp(f ), thus the following limit exists and is contin∂x uous Z ∂f 1 f (z + h + w) − f (z + w) 1 lim dw ∧ dw¯ = (z + w) dw ∧ dw¯ h∈R,h→0 C h w ∂x w
This implies To verify
∂u ∂ z¯
∂u ∂x
exists and is continuous.
= f , we write ∂u 1 = lim ∂ z¯ 2πi →0
Z |w|≥
∂f 1 (z + w) dw ∧ dw¯ ∂ z¯ w
And we note Z Z Z ∂f 1 ∂ f (z + w) f (z + w)dw I= (z+w) dw∧dw¯ = dw∧dw¯ = − d ¯ w ¯ w w |w|≥ ∂ z |w|≥ ∂ w |w|≥ By Stoke’s theorem, it follows, Z Z f (z + w) f (z + w) − f (z) I= dw = 2πi · f (z) + dw w w |w|= |w|= 34
(z) Since f (z+w)−f is bounded in w, the second term above vanishes as → 0. The w result follows.
If now f ∈ C ∞ , we apply this special case to the function ϕ · f where ϕ ∈ C ∞ , taking zero outside U and one on a compact set K ⊂ U . We obtain a function u ∈ C ∞ (U ) = f on K. To complete the proof, we need Runge’s theorem in the following with ∂u ∂ z¯ form: Let U be open in C and let K ⊂ U be compact. Let L be the union of K with those connected components of U − K which are relatively compact in U . Then L is compact with the following property: any function holomorphic in a neighborhood of L can be approximated, uniformly on L, by functions holomorphic on U . ˚n+1 , Return to the proof, let {Kn }n≥1 be a sequence of compact sets in U with Kn ⊂ K ∪Kn = U such that U − Kn has no connected component relatively compact in U . n = f on a neighborhood of Kn . Then Let f ∈ C ∞ and let un ∈ C ∞ be such that ∂u ∂ z¯ un+1 − un is holomorphic on a neighborhood of Kn , so that there is hn , holomorphic on U with |un+1 − un − hn | < 2−n on Kn . P Define u := un + m≥n (um+1 − um − hm ) − h1 − · · · − hn−1 on Kn ; the series converges uniformly on Kn . We have X u = un + (un+1 − un − hn ) + (um+1 − um − hm ) − h1 − · · · − hn−1 m≥n+1
X
=u−n+1+
(um+1 − um − hm ) − h1 − · · · − hn
m≥n+1
P so that this defines a function on U . Since m≥n+1 (um+1 − um − hm ) is holomorphic ˚n+1 ⊃ Kn and ∂un+1 = f on K, we have ∂u = f on Kn for all n, i.e., on U . on K ∂ z¯ ∂ z¯ Proof of the theorem: It suffices to show H 1 (V, O) = 0 for any open covering V := {Vi }i∈I of U . Let (ψi ) be a partition of unity subordinate to {V } with Supp(ψ) ⊂ Vi for every i ∈ I. Let (cij ) ∈ Z 1 (U, O), we define a function ψj · cij on Ui by P assigning it the value zero on Ui − Supp(ψj ), so that ψj · cij ∈ O(Ui ). Define gi := j∈I ψj · cij ∈ O(Ui ) (which is well-defined by the local finiteness of the family {Supp(ψi )}i∈I ). On Ui ∩ Uj , for every i, j ∈ I, we have X X gi −j = ψk · (cik − cjk ) = ψk · cij = cij (3.7) k∈I
k∈I
hence, on Ui ∩ Uj , ∂gi ∂gj ∂cij − = =0 ∂ z¯ ∂ z¯ ∂ z¯ i Thus, by sheaf axiom, there exists a global C ∞ function f on U with f |Ui = ∂g for ∂ z¯ ∂u ∞ every i ∈ I. Now, by our lemma, we can find some u ∈ C and ∂ z¯ = f on U . We set
35
k hk := gk − u on each Uk , then ∂h = 0 so hk ∈ O(Uk ). Meanwhile, on each Ui ∩ Uj , ∂ z¯ we have hi − hj = (gi − u) − (gj − u) = gi − gj = cij
where the last identity follow from (3.7). Therefore (cij ) ∈ B 1 (U, O), whence H 1 (U, O) = 0.
36
Chapter 4 Vector Bundles, Sections and Divisors 4.1
Vector Bundles and Sections
Let X and E be two topological spaces. A continuous map π : E → X satisfying local trivialization conditions is called a vector bundle. ∀P ∈ X, its pre-image π −1 (P ) is a fibre bundle, denoted by EP , which forms a vector space of dimension n over C. For any P ∈ X, there is an open neighborhood UP and a homeomorphism h : π −1 (UP ) → UP × Cn . This homeomorphism is called local trivialization of E on U at P , and it induces a projection map pr : U × Cn → UP . Here pr ◦ h and π are the same map restricted to UP ; in other words, the following diagram commutes locally.
h
U π −1 (U ) −−− → U × Cn π pr y y U
(4.1)
U
U
Now we define a map ϕp : Ep → Cn through local trivialization h(x) 7→ (p, ϕp (x)), ∀x ∈ Ep . This is an isomorphism of C-vector spaces between each Ep and Cn . n is defined to be the rank of the vector bundle. In particular, when n = 1, it is called a line bundle, which is of particular interest to us and we will discuss its relation to divisor in next section. Now we specify the study on holomorphic vector bundle on Riemann surface X. The vector bundle is holomorphic in the sense that the trivializations are holomorphic. 37
Let U = {Ui }i∈I be an open covering of X and on any non-empty intersection Ui ∩ Uj , n n we have a biholomorphism hi ◦ h−1 j : (Uij ) × C → (Uij ) × C : (x, v) 7→ (x, η(x, v)). Note that u = η(x, v) ∈ Cn and for a fixed x, this induces an isomorphism of Cn , which belongs to GL(n, C). Therefore, for each Ui ∩Uj , there is a holomorphic map gij from Ui ∩ Uj into GL(n, C), to each x ∈ Ui ∩ Uj , we have hi ◦ h−1 j (x, v) = (x, gij (x) · v). −1 −1 −1 If x ∈ Ui ∩ Uj ∩ Uk , we have hi ◦ hk = (hi ◦ hj ) ◦ (hj ◦ hk ), which gives the cocycle condition gij · gjk = gik on Ui ∩ Uj ∩ Uk . These {gij }’s are transition functions of the vector bundles corresponding to local trivializations. For another holomorphic trivializations {h0i }, where h0i is also associated with Ui , the composition h0i ◦h−1 i is of the form (x, v) 7→ (x, φi (x)·v) for some φi (x) ∈ GL(n, C). To see this, recall, for each y ∈ Ex with x ∈ Ui , we have hi (y) = (x, ϕx (y)) = (x, v), and h0i (y) = (x, ϕ0x (y)), where ϕx (.) and ϕ0x (.) are two isomorphisms of C-vector spaces, −1 from Ex to Cn . Thus, v ≡ ϕx (y) 7→ ϕ0x (y) is given by ϕx ◦ ϕ−1 x (v). As ϕx ◦ ϕx is an n invertible linear transformation of C which depends on x, we can be written it as φi (x) · v, for some φi (x) ∈ GL(n, C). Therefore the corresponding transition functions satisfy gij · v = gij0 φφji · v, namely, gij = gij0 · φφji . Conversely, given a family of holomorphic maps gij : Ui ∩ Uj → GL(n, C) satisfying the cocycle condition, we can construct a holomorphic vector bundle as follows. S Let U = {Ui }i∈I be the covering on X. We take the disjoint union F := i∈I Ui × Cn with the map π 0 : F → X : (x, v) 7→ x. We define the following equivalence relation: for any (x, v) ∈ Ui × Cn and (y, w) ∈ Uj × Cn , (x, v) ∼ (y, w), if and only if x = y and v = gij (x) · w. This is indeed an equivalence relation due to the cocycle relation. Taking quotient of this equivalence relation on F , we obtain a topological space E := F/ ∼ (with quotient topology) whose elements are equivalence classes, each of which represents a unique element in the fibre Ep , denoted by (p, [v]). The map 0 −1 π : E → X is well-defined and induces a local trivialization h−1 from Ui ×Cn Ui := π ◦π −1 0 to π (Ui ): Clearly it is surjective for (p, v) ∈ π ◦ π{(p, [v])}; injectivity follows from the cocycle condition, for (p, v1 ) and (p, v2 ) ∈ Ui × Cn , π 0 ◦ π −1 (v1 ) = π 0 ◦ π −1 (v2 ) if and only if v1 = gii ·v2 = v2 . It then follows that the map π : E → X is a holomorphic vector bundle. Definition(Section): If π : E → X is a vector bundle, a continuous (resp. C ∞ , holomorphic) sections is a continuous (resp. C ∞ , holomorphic) map s : X → E such that π ◦ s = identity on X. In terms of local trivializations hi : π −1 (Ui ) → U × Cn , then hi ◦ s(P ) = (P, fi (X)) for P ∈ Ui , where fi is a map Ui → Cn . Since hi ◦ s(P ) = hi ◦ h−1 j (P, fj (P )) = (P, gij (P ) · fj (P )) for any P ∈ Ui ∩ Uj , we have fi (P ) = gij (P ) · fj (P ) ∀ P ∈ Ui ∩ Uj The converse is also true. Thus a continuous (resp. C ∞ , holomorphic) section of π : 38
E → X can be identified with a family {fi }i∈I of continuous (resp. C ∞ , holomorphic) maps fi : Ui → Cn such that fi = gij · fj on the intersections. In particular, a meromorphic section for a holomorphic vector bundle π : E → X is defined as follows. Let S ⊂ X be a discrete set and let s : X − S → E be a holomorphic section. Then s is a meromorphic section of E if for any P ∈ S, there is a neighborhood U of P and a coordinate z on U with z(P ) = 0 such that U ∩ S = {P } and for some integer N ≥ 0, z N · s is the restriction to U − {P } of a holomorphic section of E over U . If U = {Ui }i∈I is an open covering of X amd hi : π −1 (Ui ) → Ui × Cn are the trivializations, then we can write hi ◦ s(P ) = (P, fi (P )) for P ∈ Ui − S; fi is holomorphic on Ui − S and the section s is meromorphic if and only if fi is meromorphic on Ui for all i, i.e., poles of fi , if exist, would only locate on Ui ∩ S. If π : E → X, π 0 : E → X are two vector bundles, a morphism u : E → E 0 is a map such that π 0 ◦ u = π and u : π P → π 0−1 (P ) if C-linear for any P ∈ X. E,E 0 are isomorphic if there are morphisms u : E → E 0 and u0 : E 0 → E such that u ◦ u0 = identity on E 0 and u0 ◦ u = identity on E. A bundle π : E → X is trivial if it is isomorphic to the trivial bundle prX : X × Cn → X where prX (P, v) = P is the projection map.
4.2
Divisors
Definition: Let X be a Riemann surface. A divisor D on X is a map D : Z → Z such that the support of D is locally finite, i.e. ∀ K ⊂ X compact, the set {P ∈ X|D(P ) 6= 0} is finite, and we usually write it as X D= nP · P (4.2) P ∈X
Here nP = D(P ) ∈ Z and the sum above is finite locally on X. If X is compact, then any open covering of X has a finite sub-cover, in each of which the local sum is finite, hence the divisor D defined by (4.2) P is also finite in the compact case. The degree of a divisor is defined by deg(D) = P ∈X nP . The sum and difference of divisors D1 ,D2 is defined by (D1 ± D2 )(P ) = D1 (P ) ± D2 (P )∀ P ∈ X. Thus the set of all divisors form an Abelian group Div(X). A divisor with nP ≥ 0∀ P ∈ X is said to be effective. This gives us a partial ordering in Div(X): D1 ≥ D2 if and only if D1 − D2 is effective. Definition: Let f be a meromorphic function on X, P1 , · · · , Pk be its zeros with multiplications M1 , · · · , Mk and Q1 , · · · , Qn be its poles with multiplication N1 , · · · , Nn . 39
P P The divisor D = ki=1 Mi · Pi − nj=1 Nj · Qj is called the divisor of f , denoted by (f ). A divisor is called principal if ∃f meromorphic on X such that (f ) = D. Clearly we have (f · g) = (f ) + (g) where on the left hand side the multiplication is between two meromorphic functions while the addition on the right hand side is the sum of divisors. Two divisors D1 and D2 are called linear equivalent if the divisor D1 − D2 is principal, and the corresponding equivalence class is called the divisor class. Now, let s be a meromorphic section of a holomorphic vector bundle E on X and s 6≡ 0. For a point p ∈ X, we choose a coordinate neighborhood (U, z) with z(p) = 0 and a local trivialization h : E|U → U × Cn , then h ◦ s(x) = (x, f (x)), x ∈ U and f is an n-tuple of meromorphic functions (f1 , f2 , · · · , fn ) on U . We define the order of section s to be the integer k such that gi = fi /z k is holomorphic for all i on U , and g := (g1 , g2 , · · · , gn ) 6= (0, 0, · · · , 0). We denote this k by ordP (s). Thus we can define the divisor of a meromorphic section in the fashion for meromorphic functions: X (s) = ordP (s) · P (4.3) p∈X
Conversely, any divisor on a Riemann surface can be constructed as the divisor of a meromorphic section, in the following way. Let X be Riemann surface and D = P P ∈X nP · P be a divisor on X, which is locally finite. Let S be the support of D and (UP , zP ) be a local coordinate system at P ∈ S with zP (P ) = 0. It is always possible for us to choose UP ’s so that UP ∩ UQ = ∅ for any P 6= Q and in S. On the set U0 = X − S we define f0 ≡ 1, and let fP = zPnP on UP , ∀ P ∈ S. Now we consider the transition function gij = fi /fj on Ui ∩ Uj 6= ∅. The only situations is that Ui = U0 while Uj = UP for some P ∈ S. Then the transition functions are gij = 1/zPnP and gji = zPnP , both are holomorphic and nowhere zero on the intersection. Moreover, on any Ui ∩ Uj ∩ Uk , we only need to consider the case where Ui = Uk = U0 and Uj = UP for some P ∈ UP , then we have gij = 1/zPnP and gjk = zPnP , thus gij · gjk = 1 = gik . Hence the cocycle condition is also satisfied. The collection of transition functions {gij } define a holomorphic line bundle L[D] on X and {fi } define the corresponding meromorphic section sD . Now we see, for any point Q ∈ X − S, ordQ (sD ) = ordQ (f0 ) = 0 and for any Q = P ∈ S, ordQ (SD ) = ordP (zPnP ) = nP . Therefore (sD ) = D. If we use a different local coordinate (UP , ζP ) at P , then hP = (ζP /zP )( nP ) is holomorphic and non-zero on UP . If we set h0 ≡ 1 on U0 = X − S and denote by {gij0 } the transition functions obtained from the functions ζPnP , then gij0 = hi · gij · h−1 j . If 0 0 L is the line bundle defined by the {gij }, there is an isomorphism of L[D] which we defined above onto L0 taking sD to the the corresponding section of L0 defined by {fi0 } (where f00 = 1, fP0 = ζPnP ). As a general remark, we note that if s is a non-zero meromorphic sections of a holomorphic vector bundle on X, then s is holomorphic is and only if the divisor associated to s is effective. 40
Given a divisor D and an open subset U ⊂ X, let OD (U ) = {f ∈ M(U ) | (f ) ≥ −D}. The assignment U 7→ OD (U ) is clearly a sheaf, denoted by OD . If Γ(X, L[D]) denotes the space of holomorphic sections of L[D] over U , and sD ∈ Γ(X, L[D]) is the standard section with (sD ) = D. The map OD (U ) → Γ(X, L[D]) with f 7→ f · sD is an isomorphism. Thus the sheaf of germs of holomorphic sections of L[D] can be canonically identified with OD . If π : L → X is a line bundle, and let s0 , s0 be two meromorphic sections of L with s0 6≡ 0, then there is a meromorphic function f on X with s1 = f · s0 . Definition(Linear Equivalence): Two divisors D1 , D2 are said to be linearly equivalent if there exists a non-trivial meromorphic function f on X such that (f ) = D1 −D2 . Lemma: Two divisors D1 , D2 are linearly equivalent if and only if the line bundles L[D1 ] and L[D2 ] are holomorphically isomorphic. Proof: Suppose D1 and D2 are linearly equivalent and let f ∈ M(X) such that (f ) = D1 − D2 . Let sD1 and sD2 be the standard sections of L[D1 ] and L[D2 ], there is a unique isomorphism u : L[D1 ] → L[D2 ] taking sD1 to f · sD2 , defined at P ∈ X outside the support of (f ), D1 and D2 by λ · sD1 (P ) 7→ λ · f (P ) · sD2 (P ) λ ∈ C
(4.4)
It can be extended holomorphically to X for on any open subset U ⊂ X containing a point P , the section P 7→ λ(P ) · sD1 (P ) is holomorphic if and only if P 7→ λ(P ) · f (P ) · sD2 (P ) is holomorphic. Conversely, if u : L[D1 ] → L[D2 ] is an isomorphism, then u ◦ sD1 is a meromorphic section of L[D2 ]. If f is defied by u ◦ sD1 = f · sD2 , we have (f ) = D1 − D2 since ordP (sD1 ) = ordP (u ◦ sD1 ) for any P ∈ X.
41
Chapter 5 Differential and Integral on Riemann Surfaces 5.1
Differential Forms
Let X be a Riemann surface, for any open subset Y ⊂ X, we consider a function f : Y → C such that for every coordinate chart (U, z) with U ⊂ Y and z : U → V ⊂ C, there exists a function f˜ ∈ C ∞ (V ) and f |U = f˜ ◦ z (from now on, we will use f instead of f˜ as long as this is not misleading). Here, C ∞ (V ) is the set of all complex valued functions on V ⊂ C ∼ = R2 and are infinitely differentiable with respect to real coordinates. It can be shown C ∞ (V ) is an algebra over C. The set of such functions with the natural restriction form a sheaf of differentiable functions on X. For a point a ∈ X, the stalk of all germs of differentiable functions at a is denoted by Ea ; the subspace ma ⊂ εa consists of germs that vanish at a and m2a ⊂ ma is the subspace of germs that vanish to second order, i.e. for a coordinate chart (U, z) containing a, with z(a) = 0, ∂f /∂z = ∂f /∂ z¯ = 0 at a, for every f ∈ ma . These definitions are independent of the choice of coordinates, for, suppose we have a smooth complex-valued function f on U ⊂ X, with two local coordinate charts (Uα , zα ) and (Uβ , zβ ), where Uα and Uβ are subsets of U and their intersection is non-empty. By definition, we have f |Uα = f˜|Uα ◦ zα and f |Uβ = f˜|Uβ ◦ zβ . Clearly, on Uα ∩ Uβ , we have f˜|Uα ◦ zα = f˜|Uβ ◦ zβ , therefore, f˜|Uα = f˜|Uβ ◦ ϕ where ϕ = zβ ◦ zα−1 is the transition function, which is holomorphic and non-vanishing on the intersection. Thus, in local coordinates, ∂ f˜|Uα /∂zα = 0 is equivalent to ∂ f˜|Uβ /∂zβ = 0, so is it ¯ with ∂-operator. (1)
Definition: The quotient vector space Ta = 43
ma , m2a
called cotangent space of X at a,
is a C-vector space of differentials at a, denoted by da . For any differentiable function f in a neighborhood U of a, it is defined as da (f ) = (f − f (a)) mod m2a With respect to local coordinates z = x + iy, we can write da f =
∂f ∂f ∂f ∂f (a)da x + (a)da y = (a)da z + (a)da z¯ ∂x ∂y ∂z ∂ z¯
Indeed the cotangent space is spanned by the basis (da , dy ) over C. To see this, we take some ϕ ∈ ma representing an equivalence class [ϕ] in the cotangent space. The Taylor expansion of ϕ at a is ϕ = c1 (x − x(a)) + c2 (y − y(a)) + ψ, where ψ ∈ m2a , thus [ϕ] = c1 · dx + c2 · dy . Meanwhile, c1 · dx + c2 · dy = 0 implies ∂{c1 · (x − x(a)) + c2 · (y − y(a))} = c1 = 0 ∂x and
∂{c1 · (x − x(a)) + c2 · (y − y(a))} = c2 = 0 ∂y
Therefore, da x, da y are linearly independent. Now for any differentiable function f in a neighborhood of a, by Taylor expansion, we have f − f (a) =
∂f ∂f (a)(x − x(a)) + (a)(y − y(a)) + g ∂x ∂y
where g ∈ m2a . By definition, we have da f =
∂f ∂f (a)da x + (a)da y ∂x ∂y
Similarly, we see that the pair (da z, da z¯) also forms a basis for the cotangent space, through the transformation dx =
dz + d¯ z dz + d¯ z and dy = 2 2i
Basically, for two coordinate charts (U, z) and (V, w) that both contain a, we want to know whether differentials of a smooth function under two basis (da z, da z¯) and (da w, da w) ¯ are the same. On the intersection U ∩ V , we have w = φ ◦ z where φ is the transition function. By definition, da w = (w − w(a))mod m2a . With respect to (U, z) we have, for some c ∈ C, da w − c · da z = (φ ◦ z − φ ◦ z(a)) − c(z − z(a)) mod m2a 44
As claimed before, m2a is independent of the choice of coordinates, hence the operations on equivalence classes make sense without specifying a coordinate chart. Clearly, ∂{(φ ◦ z − φ ◦ z(a)) − c(z − z(a))} =0 ∂ z¯ while, ∂φ ∂{(φ ◦ z − φ ◦ z(a)) − c(z − z(a))} = −c=0 ∂z ∂z (1)
¯
(1,0)
(0,1)
precisely when c = ∂φ . In other words, da w = ∂φ d z in Ta . Similarly, da w¯ = ∂∂φz¯ da z¯. ∂z ∂z a Now, for a smooth complex-valued function whose domain contains U ∩ V , let f , g be its local expressions in (U, z) and (V, w) respectively, we have f = g ◦ φ on the ∂g ∂g ∂g intersection, hence ∂f = ∂w · ∂φ and ∂f · da z = ∂w · ( ∂φ ) · ( ∂φ )−1 · da w = ∂w · da w. ∂z ∂z ∂z ∂z ∂z ∂f ∂g Similarly, we have, ∂ z¯ · da z¯ = ∂ w¯ · da w. ¯ Therefore da f is independent of the choice of coordinates. (1)
In particular, we can decompose Ta into direct sum of Ta := C ⊗ da z and Ta := C ⊗ da z¯. Here Ta (1, 0) and Ta (0, 1) are called cotangent space of type (1, 0) and type (0, 1) respectively. S (1) A differential 1-form on U ⊂ X is defined as a map ω : U → a∈U Ta . To each point (1) a ∈ U , it assigns an element ω(a) ∈ Ta . We may write it as ω = p(z, z¯)dz + q(z, z¯)d¯ z in local coordinate (U, z), where p(z, z¯),q(z, z¯) are complex valued functions, but are not necessarily smooth or continuous in general. When we talk about smooth differential 1-form, we further require they be smooth. This definition is also independent d¯ z2 dz2 and q(z1 , z¯1 ) = q(z2 , z¯2 ) d¯ on the interof local coordinates: p(z1 , z¯1 ) = p(z2 , z¯2 ) dz z1 1 section of two sub-coordinate charts (U1 , z1 ), (U2 , z2 ). An 1-form is called a (1, 0)-form if ω = p(z, z¯)dz and a (0, 1)-form if it is written as ω = q(z, z¯)d¯ z . In particular, a (1, 0)-form is holomorphic if p(z, z¯) is holomorphic (we may write it as a single variable function in z only). Similarly, a (0, 1)-form with q(z, z¯) being antiholomorphic is called an antiholomorphic 1-form. Suppose there is a point a ∈ X. We choose a coordinate chart (U, z) so that z(a) = 0, then an 1-form ω holomorphic on some open neighborhood YP⊃ U excluding a can n be locally written as f dz where f has a Laurent expansion ∞ n=−∞ cn z . Clearly, −1 f (or more precisely f ◦ z ) is a holomorphic function on the punctured domain U − {a}. In the Laurent expansion, if for all n < 0 cn = 0, ω can be holomorphically continued to Y and the point a is a removable singularity; If for some k < 0, ck 6= 0 and for all n < k cn = 0, ω is a pole of kth order at a; Otherwise, ω is an essential singularity. The coefficient c−1 is called the residue of ω at a, which is independent of the To see this, we note that any 1-form in form of dg where Pchoice of coordinates. n g= ∞ c z is a holomorphic function in some punctured neighborhood V − {a} n=−∞ n of a has 0 as the coefficient for the term z −1 dz, which is independent of the choice 45
of coordinates; Meanwhile, for any coordinate chart (U, z) with z(a) = 0, suppose ϕ = z · h where h is a holomorphic function on U and h(0) 6= 0, i.e., h does not = 1, which is independent of the choice of vanish at a, then the residue at a for dϕ ϕ the coordinates again. Now with this two facts, we pick chart (U, z) P up a coordinate n and write a holomorphic function f on U − {a} as ∞ c z , it can be shown n n=−∞ P−2 P∞ cn n+1 cn n+1 −1 ω = f dz = dg + c−1 z dz, where g = + n=0 n+1 z is clearly n=−∞ n+1 z holomorphic on U − {a} and z is the ϕ we used above. Hence the residue of ω at a is indeed c−1 and is unchanged if we reparametrize ω. With these definitions, we can give a formal definition of meromorphic 1-form on Riemann surface, and we shall discuss it in detail in the next section focusing on the integral of differential forms. Definition:A meromorphic 1-form µ on some open set Y ⊂ X is holomorphic on Y except for a set of isolated points, at which, µ has poles. The set of all meromorphic 1-forms on Y form a sheaf of vector space, denoted by M(1) (Y ). (2)
Second order cotangent space at a, denoted by Ta is the space of 2nd order alter(1) nating tensors on space Ta . This is one-dimensional space with basis da z ∧ da z¯ = −2ida x ∧ da y. Similarly, for an open set Y ⊂ X, a 2-form ω is a map that assigns (2) to each point a ∈ Y an element in Ta . In every coordinate chart (U, z), we can write it as ω = f dz ∧ d¯ z , and it is differentiable if f is differentiable. Again, this definition shall be independent of the choice of coordinates, recall for f : Y → C, ∂ϕ dz2 and d¯ z1 = ∂∂ϕ d¯ z2 where ϕ = z1 ◦ z2−1 , we have locally f2 = f1 ◦ ϕ, dz1 = ∂z z¯2 2 ∂ϕ 2 | dz2 ∧ d¯ z2 = f2 dz2 ∧ d¯ z2 . Therefore we must have hence, f1 dz1 ∧ d¯ z1 = f1 · | ∂z 2 ∂ϕ 2 0 2 f1 = f2 · | ∂z2 | = f2 · |z1 (z2 ) | . The differentiation d of forms are linear operations with Leibniz rule, sending differential n-forms to (n + 1)-forms. In particular, in the case that n = 0, i.e., differentiation on 0-forms, namely, smooth functions, it matches our previous definition. In the case n ≥ 1, we introduce the exterior differentiation. For n = 1, we have, following Leib+ ∂g )dz ∧ d¯ z . For n = 2, niz rule, dω = d(f dz + gd¯ z ) = df ∧ dz + dg ∧ d¯ z = (− ∂f ∂ z¯ ∂z dω = d(f dz ∧ d¯ z ) = 0. Definition: A smooth differential 1-form ω on some open set Y ⊂ X is called closed if dω = 0 and is exact if ω = df for some differentiable function f on Y . 2
2
∂ f ∂ f Clear, ddf = d( ∂f dz + ∂f d¯ z ) = ( ∂z∂ − ∂z∂ )dz ∧ d¯ z = 0, hence every exact form is ∂z ∂ z¯ z¯ z¯ closed. The converse is not true in general (we will see that a closed 1-form has a primitive hence is exact locally). It follows that every holomorphic or antiholomorphic 1-form is closed and conversely, every closed 1-form of type (1, 0) is holomorphic and every closed form of type (0, 1) is antiholomorphic.
The notion of closed and exact forms can be extended to higher order forms which induces the De-Rham Cohomology:
46
Definition: Let Ωk be the space of closed differential k-forms (an abelian group) and dΩk−1 be the space of exact k-forms; Let Z k be the space of closed k-forms. Then dΩ(k−1) is a normal subgroup of Z k . Taking the quotient, we have the k-th de-Rham cohomology group: Zk (5.1) H k (X, C) := dΩ(k−1)
5.2
Integration of Differential Forms
The integration of differential k-forms is carried out on k-chains, which we discussed earlier. For example, say c0 = R for aP0-form, namely a function f and a 0-chain, P Pn ki f (Pi ); For an 1-form ω and a 1-chain c1 = ni=1 ki γi i=1 ki Pi , we have c0 f = (As we have seen before, homology is in fact independent of the triangulation, the definitionR of 1-chain R be generalized to finite union of smooth oriented curves), P can we have c1 ω = ki γi ω, where the integrals on the right hand sides are curve integrals. For each smooth curve we may partition it into local coordinate charts and integral piecewisely, i.e. a smooth curve γ(t) : [0, 1] → X can be partitioned into 0 = t0 < t1 < t2 < · · · < tn = 1 such that for each interval γ([tk−1 , tk ]) ⊂ Uk , with a collection of coordinate charts (Uk , zk ), k = 1, 2, ..., n. In each chart ω = fk dxk +gk dyk , so that the integral along γ is: Z ω := γ
n Z X k=1
tk
tk−1
{fk (γ(t))
dyk (γ(t)) dxk (γ(t)) + gk (γ(t)) }dt dt dt
(5.2)
Since differential forms are independent of the choice of coordinates, this definition is independent of the partition of curves and hence makes sense. One can easily see that if ω is globally an exact 1-form, say ω = dF for some smooth function F on X (i.e.F is the primitive for ω), then an integral along a piecewisely smooth curve R only depends on the initial and the end points, namely, we have γ dF = F (b) − F (a) where γ(0) = a and γ(1) = b. Similarly, integral of 2-forms are defined for 2-chains, which Pnare generalized to finite union of domains, i.e. for a 2-form ω and a 2-chain Rc2 = P , where every Di i=1 ki Di R is a domain on the Riemann surface, the integral is c2 ω = ni=1 ki Di ω. For a smooth 1-form ω, its integral on a closed piecewisely smooth curve given by (5.2) can be re-written as: Z ω= γ
n Z X k=1
n
tk
X ∂gk (γ(t)) ∂fk (γ(t)) dxk ∧ dyk + dxk ∧ dyk } = {− ∂dyk ∂dxk tk−1 k=1
Z dω
(5.3)
Dk
S where ni=1 Dk is the domain bounded by the curve γ and the last identity follows from Green’s Theorem. 47
As a generalization, this gives us the Stokes’ theorem for differential forms: Theorem: Let D be a 2-chain with a piecewisely smooth boundary ∂D, then the formula Z Z dω = ω (5.4) D
∂D
holds for any differential forms ω. Applying Stokes’ theorem to 1-forms, we have two important results: Integral of close forms on a cycle is independent of its representation in de Rham cohomology, for two 1-forms ω1 and ω2 = ω2 + dF , Z Z Z Z ω2 = ω1 + dF = ω1 + f (γ(1)) − f (γ(0)) = ω2 (5.5) γ
γ
γ
γ
Meanwhile, integral of a closed 1-form does not depend on the representation in the homology class, for two homologous cycles γ1 and γ2 , Z Z Z Z dω = 0 (5.6) ω= ω= ω− γ2
γ1
∂D
D
Here D is the domain bounded by γ1 and γ2 , since 0 = γ1 −γ2 = ∂D for some 2-chain.
5.3
Abelian differentials of three kinds
In this section, the discussion will be in the context of compact Riemann surface. Recall the simply connected 4g-gon model for a compact Riemann surface X with genus g, which is denoted by ∆ as before. a1 , b1 , · · · , ag , bg is the basis for H1 (X, Z). For a closed R P differential 1-form ω, we fix a point P0 in ∆, and define the function u(P ) = P0 ω (this notation will be used in the whole section), ∀P ∈ ∆ and the RP integration path lies in ∆. It follows that d( P0 ω) = ω(P ). For a closed R 1-form ω inR a neighborhood of these curves a1 , b1 , · · · , ag , bg , we define Ai (ω) = ai ω, Bi (ω) = bi ω, which are called a− and b− periods for ω respectively. P P Any closed curve γ on X is homologous to Rgi=1 ni aiP + gi=1 mi biP for some integervalued mi , ni , i = 1, 2, · · · , g. Hence, by (5.6), γ ω = gi=1 ni Ai + gi=1 mi Bi . Lemma: Let ω beSa closedS1-form on X, ϕ be a smooth closed 1-form defined in a neighborhood gi=1 ai ∪ gj=1 bj , we identify them with 1-forms on ∆ and ∂∆ respectively. It follows, Z
Z u(p)ϕ =
∂∆
∂∆
Z (
p
p0
ω)ϕ =
g X k=1
48
{Ak (ω)Bk (ϕ) − Bk (ω)Ak (ϕ)}
(5.7)
Proof: P is a point on the boundary of ∆, without loss of generality, we assume P ∈ ak RP 0 . Then u(P ) − u(P ) = α along a and let P 0 be the corresponding point on a−1 k P0 −1 0 curve γ from P to P , which is homologous to bk . Since ω is closed, the integral is R 0 independent of the representation in homology class, hence u(P ) − u(P ) = b−1 α = k R 0 −Bk (ω). Similarly,for Q ∈ bk and Q0 ∈ b−1 , we have u(Q) − u(Q ) = ω = A k (ω), k ak therefore we have, Z g Z X uϕ = ( +
Z ∂∆
=
g Z X k=1
k=1
Z +
+
a−1 k
ak
0
(u(P ) − u(P ))ϕ(P ) +
ak
Z
bk
g Z X k=1
)uϕ b−1 k
(u(Q) − u(Q0 ))ϕ(Q)
bk
Z Z g X ϕ ϕ + Ak (ω) = (−Bk (ω) k=1
=
g X
ak
bk
{Ak (ω)Bk (ϕ) − Bk (ω)Ak (ϕ)}
k=1
In particular, let ω be holomorphic, and Rwe may choose ¯ be the closed forms R R ϕ=ω above, then by Stokes Theorem, ∂∆ u¯ ω = ∆ du ∧ ω ¯ = Xω∧ω ¯ . In a localRcoordinate (U, f ∈ O(U ), so that U ω ∧ ω ¯ = R z),2 we have ω R= f2dz for some homomorphic R 1 |f | z ∧ z¯ = −2i |f | dx ∧ dy. Therefore 2i X ω ∧ ω ¯ ≤ 0. Substituting into (5.7), U we have Z g g X 1 X 1 ω∧ω ¯= {Ak (ω)Bk (ω) − Bk (ω)Ak (ω)} = Im Ak (ω)Bk (ω) ≤ 0 2i X 2i k=1 k=1 And the equality is attained only when ω = 0. As a result, if all the a− periods for holomorphic ω are zero, ω must be zero. A holomorphic differential 1-form ω on a compact Riemann surface X is called abelian differential of the first kind. Holomorphic differential 1-forms form a complex vector space, denoted by H 0 (X, Ω), and dimH 0 (X, Ω) ≤ g. To see R this, suppose we have j g + 1 holomorphic 1-forms, ω1 , · · · , ωg+1 ; denote by Ai for aj ωi , the linear system Pg+1 j · · · , g has a non-trivial solution (α1 , · · · , αg+1 ), hence all i=1 αi Ai = 0 for j = 1, P2, g+1 the a− periods for ω = i=1 αi ωi are zero, which implies ω = 0, i.e. (omegai )1≤i≤g+1 are linearly dependent. In fact, the dimension of the space of holomorphic 1-forms H 0 (X, Ω) of a compact Riemann surface X is equal to the genus g, which will be proved later using RiemannRoch theorem and Serre duality (Indeed, by Serre duality theorem, dimH 0 (X, Ω) = dimH 1 (X, O(X)), and by Riemann-Roch theorem, dimH 1 (X, O(X)) = dimH 0 (X, O(X))− 49
1 + g; since H 0 (X, O(X)) ∼ = O(X), which consists only constant functions on a compact Riemann surface, we have dimH 0 (X, O(X)) = 1, and the result follows). Assuming this fact, we take aRbasis ω1 , · · · , ωg for H 0 (X, Ω), from above, we see the matrix (Aji )ij where Aji = ai ωj is invertible. In view of this, we can choose the basis so that Aji = δij , and the basis is called normalized with respect to (ai , bi ), for i = 1, · · · , g. With this normalized basis, we have the following bilinear relation: j j RTheorem(Riemann’s bilinear relation): The complex matrix B = (Bi )ij where Bi = ω , is symmetric with its imaginary part being positive definite. bi j R RP R Proof: Let uj (P ) = P0 ωj , by Stokes’ theorem, we have ∂∆ uj ωk = X ωj ∧ ωk = 0, for locally, ωi = fi dz, and ωj = fj dz, thus their exterior products vanish in (U, z). On the other hand, from the previous lemma, we have
Z ωk = ∂∆uj
g X
{Ai (ωj )Bi (ωk ) − Bi (ωj )Ai (ωk ) = Bj (ωk ) − Bk (ωj ) for Ai (ωj ) = δij
i=1
Pg Thus, B is symmetric. Now, let v = (c1 , · · · , cg ) 6= 0 ∈ Rg , and ω = i=1 ci ωi , P P P Pg we have Im i=1 Ai ( ck ωk )Bi ( ck ωk ) < 0. Since Ai ( ck ωk ) = ck , we have P P Im i,k ci ck Bik < 0, i.e. Im i,k ci ck Bik = v T · ImB · v > 0. Recall the definition of meromorphic 1-form and residue. The coordinate independence of the definition of residue hasRbeen proved earlier. For µ ∈ M(∞) (X), residue 1 at a point P can be calculated 2πi µ for a closed simple curve around p in the γ positive direction. Let S be the set of singularities of µ, by definition, they form a discrete set of poles. In particular, if the Riemann surface is compact, the set S is finite, for otherwise, accumulation points exist, which become essential singularities. In the compact case, the sum of the residue of µ is zero. To see this, recall a compact Riemann surface can be modeled as 4g-gon ∆, hence can be triangulated so that poles are not on any edges of the triangles. Since µ is holomorphic on X − S, the sum of residue is equal to the sum of integrations of µ around each triangle. In the sum, µ has been integrated twice along every edge opposite R interior to ∆, oncePing each of the two−1 1 directions, and the remaining is 2πi ∂∆ µ. Since ∂∆ = i=1 = ai + bi + ai + b−1 i , the integral vanishes. Alternatively, we can prove this in the following way: Let P1 , · · · , Pm be the set of poles of µ. We choose local coordinates (Uj , zj ) about Pj with zj (Pj ) = 0 and let ∆j = {x ∈ Uj : |zj (x)| < } for some small > 0. Let 50
U =X−
Sm
j=1
∆j , by Stokes’ theorem, we have, Z XZ X dµ = − µ = −2πi (µ) U
∂∆j
j
Pj
Since, µ is holomorphic in U , it is closed, therefore the sum above vanishes. In particular, applying this to µ = dφ/φ for a non-constant meromorphic function φ on X, we find that φ has as many poles as zeros counting multiplicity, and writing φ − c for φ where x ∈ C, we find that φ takes value c as many times as it has poles. Since c is arbitrary, φ takes every value the same number of times, which is called the valence of φ. A meromorphic 1-form with singularities is called abelian differential of the second kind if the residue is zero at every pole and is of third kind if residues are non-zero. An abelian differential of second kind holomorphic in X −{P } for some P ∈ X can be (N ) locally written as µP = zNdz+1 + ωP in (U, z) with z(P ) = 0 where ωP is holomorphic at P . Similarly, an abelian differential of third kind has a pair of singularities at R R + ωR in (UR , zR ) and Q, with ResR (µ) = 1, and ResQ (µ) = −1. Locally, µRQ = dz zR dz
near R with zR (R) = 0 and µRQ = zQQ + ωQ in (UQ , zQ ) near Q with zQ (Q) = 0. And we may add proper linear combinations ofR ω1 , · · · , ω1 to meromorphic 1-forms preserving residues, so that all the a− periods ai µ vanish. Such 1-forms are called normalized abelian differentials of second kind and of third kind respectively. It is important that any meromorphic 1-form on X is a unique linear combination of these three kinds of 1-forms. The uniqueness is clear, for the holomorphic difference of two normalized differentials with the same singularities has all zero a− periods hence vanishes identically. Theorem(Reciprocity): Let ω1 , · · · , ωn be a normalized basis of H 0 (X, Ω), and let (n) µP , µRQ be normalized abelian differentials of second and third kind respectively. WeRhave, RR (1) bk µRQ = 2πi Q ωk where the integral on the right hand side is taken along a curve joining R and Q in X − ∪ai − ∪bi ; R (n) P i (2)In the local coordinate (U, z), we may write ωk = ∞ i=0 ck,i z dz, then bk µP = 2πi c . n k,n−1 Proof: Since µRQ is normalized, all its a− periods are zero while. In (5.7), since µRQ is holomorphic in X −{R}−{Q}, we can substitute and ωk into ω. Setting Rx R µRQ into ϕP uk (x) = P0 ωk for a fixed point P0 ∈ ∆, we have ∂∆ uk µRQ = ni=1 {Ai (ωk )Bi (µRQ )− R P Bi (ωk )Ai (µRQR)} = ni=1 Ai (ωk )Bi (µRQ ). Since Ai (ωk ) = δik , we have ∂∆ uk µRQ = Bk (µRQ ) = bk µRQ . On the other hand, since ∆ is simply connected and µRQ R has residue +1 at R and residue −1 at Q, residue theorem implies ∂∆ uk µRQ = RR 2πi(uk (R) − uk (Q)) = 2π Q ωk . This proves the first assertion. R (n) R (n) (n) For (2), similarly, we have bk µP = ∂∆ uk ωP = 2πi · ResP (uk µP ) = 2πi · ResP (uk · 51
dz ). z n+1
Note that
dz z n+1
= d(− n·z1 n ), hence we have,
dz 2πi · ResP (uk · n+1 ) z Z 1 = uk · d( ) (γ is a simple loop around P ) n · zn γ Z 1 =− − duk (z) (integration by part) n · zn γ Z ∞ X 1 = { × ck,i z i dz} n n · z γ i=0 Z ∞ X 1 2πi ck,n−1 = ck,i+n dz = n γ n i=−n
52
Chapter 6 Meromorphic Functions on Compact Riemann Surfaces In this chapter, we will focus on meromorphic functions on compact Riemann surfaces. We have seen that compact Riemann surfaces can be classified by their genera g, namely, when g = 0, a compact Riemann surface X is isomorphic to the complex ˆ when g = 1, it is isomorphic to the complex torus C/Λ, where Λ is a lattice sphere C; in C whose definition will be formalized in the first section of this chapter; if g ≥ 2, it is isomorphic to ∆/Γ where ∆ is the unit disk and Γ is the Fuchsian group. In the first case, all meromorphic functions on P1 are rational functions; in the second case, a meromorphic function, if exists, must be doubly periodic, which reduces our study to elliptic functions; in the last case, we will introduce the device called Poincar´e series for finding meromorphic functions. However, we shall note that this case by case study will not be generalized to the study of meromorphic functions on higher dimensional complex manifolds. In particular, for a complex torus T = Cn /Λ, which is a compact complex manifold viewed as a generalization to the genus-one compact Riemann surface, the method we will use later in this chapter by constructing elliptic functions (i.e. doubly periodic functions of one complex variable) no longer works. We will introduce Riemann forms for this particular object in Chapter 8 in the context of abelian variety. Back to the compact Riemann surface X, since a non-meromorphic function on X must attain every value as many times as the number of poles (counting multiplicity) it has, a principal divisor must be of degree zero. Abel map, which will be introduced in the next chapter provides us with a powerful device to characterize meromorphic functions on X in terms of their zeros and poles, though it does not help us to construct such functions explicitly.
53
6.1
The Riemann-Roch Theorem and Applications
As a central theorem in the study of compact Riemann surfaces, Riemann-Roch theorem tells us how many linearly independent meromorphic functions there are with certain restrictions on their poles. And in this section X will be used to denote Riemann surfaces which are compact. We first introduce a sheaf OD for a divisor D on a Riemann surface X: OD (U ) := {f ∈ M(U ) : (f ) ≥ −D on U } Together with the natural restriction maps, OD is a sheaf. If D,D0 are two linearly equivalent divisors, the OD and OD0 are isomorphic through the following map, OD → OD0 : f 7→ ψf for some non-trivial ψ ∈ M(X) such that D0 − D = (ψ).
Denote by H 0 (X, OD ) the global section of OD as a vector space over C and by l(D) its dimension . We also introduce the sheaf ΩD of abelian differentials for a divisor D such that on any open set U ⊂ X, ∀ ω ∈ ΩD , we have ordP (ω) ≥ −D(P ) for any P ∈ U , i.e., (Ω) ≥ −D. And we denote by H 0 (X, ΩD ) its global section with i(D) the associate dimension. This dimension is called index of speciality, which we will discuss later. It is obvious that D1 ≥ D2 implies H 0 (X, OD2 ) ⊂ H 0 (X, OD1 ) and l(D2 ) ≤ l(D1 ). For a non-trivial effective divisor D, we have l(D) = 0. In particular, for the zero divisor, the H 0 (X, O0 ) only contains constant functions, for otherwise, a non-constant function would take every value as many time as poles it has, which would be zero. Therefore l(0) = 1. On the other hand, H 0 (X, Ω0 ) contains all holomorphic differentials, hence i(0) = g. Since OD1 and OD2 are isomorphic provided that D1 and D2 are linearly equivalent, we have l(D1 ) = l(D2 ), i.e. l(D) depends only on the divisor class of D. Similarly, ∀ ω ∈ H 0 (X, ΩD ), let ω0 be a non-zero abelian differential so that (ω0 ) is the canonical divisor KX over X, the map ω 7→ ω/ω0 is an isomorphism between H 0 (X, ΩD ) and H 0 (X, OD+C ). Hence, we have, i(D) = l(D + C)
(6.1)
This implies that i(D) also depends on the divisor class only. Theorem(Riemann-Roch): Let X be a Riemann surface of genus g and D a divisor on X, then, dimH 0 (X, OD ) = deg(D) − g + 1 + dimH 0 (X, Ω−D ) (6.2) 54
i.e., l(D) = deg(D) − g + 1 + i(−D) Proof: We start from the case when D is effective. In particular, if D = 0, the formula holds true since l(0) = 0 and i(0) = g. Now we assume all the points of the divisor have multiplicity one, namely, D = P1 + · · · + Pk (The treatment of the general case is similar except for more complicated notations). If f ∈ H 0 (X, OD ), the its differential df lies in H 0 (X, ΩD0 ), where D0 = 2D = 2P1 + · · · + 2Pk . Moreover, it lies in the subspace V := {ω | ω ∈ H 0 (X, ΩD0 ), ResPj (ω) = R (1) 0, ai ω = 0}. We note that the normalized differentials of the second kind µPj , for j = 1, 2, · · · , k, form a basis for V , hence dimV = k = deg(D). For the map d : H 0 (X, OD ) → V , the kernel consists of constant functions only, hence dim ker(d) = 1, thus we have l(D) = 1 + dim Image(d)
(6.3)
Writing explicitly, df =
k X
(1)
f j µP j
(6.4)
j=1
where fj are constants such that
R bj
df = 0 for j = 1, 2 · · · , g.
This condition is formulated in a system of g linear equations for deg(D) = k variables, therefore we immediately obtain, Image(d) ≥ deg(D)−g, thus, for any effective divisor we have, l(D) ≥ deg(D) + 1 − g (6.5) From the inequality we conclude for any effective divisor D with deg(D) = g +1, there exists a non-trivial meromorphic function in H 0 (X, OD ). For a Riemann surface of genus 0, we consider an effective divisor D with ordP (D) = 1 and ordx (D), ∀x 6= P ∈ X. The inequality implies l(D) ≥ 2, hence there is a non-trivial meromorphic function f on X with one pole, taking every value exactly once, i.e. f is a holomorphic ˆ This shows that any Riemann surface of covering from X to the complex sphere C. genus 0 is conformally equivalent to the complex sphere. Recall Reciprocity theorem, let ω1 , · · · , ωg be a normalized basis for section Pthe global n of holomorphic differentials, namely H 0 (X, Ω), written as ωi = ∞ c z dz. We n=0 j,n R (1) have bi µPj = 2πi · ci,0 (Pj ) for j = 1, 2, · · · , k. The linear system in (6.3) becomes k X
fj · ci,0 (Pj )
j=1
c1,0 (P1 ) · · · cg,0 (P1 ) .. .. .. i.e. (f1 , f2 , · · · , fk )H = 0, where H = . . . . c1,0 (Pk ) · · · cg,0 (Pk ) 55
(6.3*)
Hence dim Image(d) = dim ker(H T ) = deg(D) − rank(H). Near the points Pj , the normalized holomorphic differentials ωi have theP asymptotics ωi = (ci,0 (Pj ) + o(1))dzj , hence a (t1 , t2 , · · · , tg ) ∈ ker(H) if and only if gi=1 ti · ωi ∈ H 0 (X, Ω−D ). Hence dimH 0 (X, Ω−D ) := i(−D) = dimH = g − rank(H). Together, we obtain (6.1) for any effective divisors. Before generalizing to an arbitrary divisor we introduce the following corollaries: Corollary: The degree of canonical divisor KX is deg(KX ) = 2g − 2. ˆ At the pole Proof: In the case that g = 0, we consider the complex sphere C. 1 ˆ with z(τ∞ ) = ∞, where the parametrization is z(τ ) = , the differential τ∞ ∈ C τ has a double pole, whence deg(dz) = −2. Since degree only depends on dz = − dτ 2 τ the class of a divisor and all abelian differentials are in the same canonical class, we have deg(KX ) = −2. If g > 0, there exists a non-trivial holomorphic differential ω whose divisor (ω) = KX is clearly effective, whence Riemann-Roch theorem holds for KX . From (6.1), we have l(KX ) = i(0) = g and i(−KX ) = l(0). Together we have deg(KX ) = 2g − 2. Corollary: On a compact Riemann surface, there is no point at which all holomorphic differentials vanish simultaneously. Proof: Suppose there is a point P ∈ X at which all holomorphic functions vanish. Let D be the divisor having order 1 at P and 0 elsewhere, i.e., i(−D) = dimH 0 (X, Ω) = g. Apply Riemann-Roch theorem, l(D) = 2, whence there is a non-constant meromorphic function f with a single simple pole at P . Therefore f is a holomorphic covering ˆ whence g = 0, and from the above corollary, we see deg(KX ) = −2. In from X to C, other words, there are no holomorphic differentials on a Riemann surface of genus 0. Now we proceed to prove Riemann-Roch theorem in general case. In fact, if a divisor D is linearly equivalent to an effective divisor, Riemann-Roch theorem holds since l(D), i(D) and deg(D) only depends on the class of the divisor. Similarly, if KX − D is linearly equivalent to an effective divisor, we have, l(KX − D) = deg(KX − D) − g + 1 + i(D − KX ). Applying (6.1) twice, we have l(D) = deg(D) − g + 1 + i(−D). If l(D) > 0, there exists some meromorphic function f ∈ H 0 (X, OD ), whence (f ) + D is effective and is equivalent to D. Similarly, if i(−D) > 0, this is equivalent to that l(C) > 0, whence for some f ∈ H 0 (X, OKX −D , we have (f ) + KX − D is effective and is equivalent to KX − D. In both cases, Riemann-Roch theorem holds and we are left with the case l(D) = i(−D) = 0. We will show in this case deg(D) = g − 1. Let D be a divisor which is not effective. We represent D as the difference of two effective divisors, i.e. D = D1 − D2 and D2 6= 0. Applying (6.5), l(D1 ) ≥ deg(D) − g + 1 = deg(D) + deg(D2 ) − g + 1. If deg(D) ≥ g, then l(D1 ) ≥ deg(D2 ) + 1, i.e., the dimension of H 0 (X, OD1 ) is at 56
least deg(D2 ) + 1. We have deg(D2 ) + 1 = k + 1 linearly independent meromorphic 0 functions f1 , f2 , · · · , fk+1 in (X, OD1 ). Suppose D2 = d1 · P1 + · · · + dm · Pm PH m with di = ordPi (D1 ) and P∞ ı=1 di =n deg(D2 ) = k, local expressions of fi around a point Pj are fi = n=0 ai,n (Pj )z dz, where z(Pj ) = 0. A linear combination Pk+1 g = i=1 ci · fi lies in H 0 (X, OD1 ) and vanishes at Pj of order at least dj precisely P if k+1 0 for n = 0, 1, 2, · · · , dj − 1 and j = 1, 2, · · · , m, which forms i=1 ci · ai,n (Pj ) = Pm a linear system of j=1 dj = k equations with k + 1 variables. Obviously, this system has a non-trivial solution (c1 , c2 , · · · , ck+1 ),P therefore by linear independence of f1 , f2 , · · · , fk+1 , the meromorphic function g = k+1 i=1 ci · fi is non-trivial, lying in 0 H (X, OD ), whence l(D) > 0, a contradiction to our assumption. This shows that deg(D) ≤ g − 1. On the other hand, with i(−D) = l(KX − D) = 0, in the same way, we obtain deg(KX − D) ≤ g − 1. Since deg(KX ) = 2g − 2, we have deg(D) ≥ g − 1. Together, we conclude that deg(D) = g − 1, which completes the whole proof of Riemann-Roch theorem.
6.2
Riemann-Hurwitz Formula
In this section, we introduce Riemann-Hurwitz formula and will apply the formula to coverings of the complex sphere. Together with Jacobi inversion theorem, we will see how the study of elliptic integrals could be replaced by the study of doubly-periodic functions. First of all, we need to formalize some definitions. Theorem(Local Behavior of Holomorphic Mappings): Suppose X and Y are two Riemann surfaces and f : X → Y is a non-constant holomorphic mapping. Suppose a ∈ X and b := f (a) ∈ Y . Then there exists an integer k ≥ 1 and coordinate charts ϕ : U → V on X and ψ : U 0 → V 0 on Y with the following properties such that (1) a ∈ U and ϕ(a) = 0 with b ∈ U 0 and ψ(b) = 0; (2) f (U ) ⊂ U 0 ; (3)the map F := ψ ◦ f ◦ ϕ−1 : V → V 0 is given by F (z) = z k for all z ∈ V . Proof: We first note that there exists charts ϕ1 : U1 → V1 on X and ψ : U 0 → V 0 on Y 0 so that (1) and (2) are satisfied. We have a function F1 := ψ ◦ f ◦ ϕ−1 1 : V1 → V ⊂ C. Since f is non-constant on X, it follows f is non-constant on U1 = ϕ−1 1 (V1 ), for otherwise, by Identity theorem, f |U1 would be extended to a constant function on X. Therefore, F1 is also non-constant, with F1 (0) = 0. There exists some integer k ≥ 1 such that F1 (z) = z k g(z), where g is a holomorphic function on V1 with g(0) 6= 0. Hence there exists a neighborhood of 0 and a holomorphic function h on this neighborhood with h(0) 6= 0 such that hk = g. The correspondence z 7→ zh(z) defines a biholomorphic mapping α from an open neighborhood V2 ⊂ V1 of zero onto an open neighborhood V of zero. Now we define U be ϕ−1 1 (V2 ) ⊂ U1 and replace ϕ1 by the map ϕ : U → V by ϕ = α ◦ ϕ1 , by construction, the mapping F := ψ ◦ f ◦ ϕ−1 57
satisfies F (z) = z k . The number k above can be characterized in the following way. For every neighborhood U0 of point a ∈ X, there exists neighborhoods U ⊂ U0 of a and W of b = f (a) such that ∀ y 6= b ∈ W , f −1 (y) ∩ U contains exactly k points. And this k is defined to be the multiplicity with which the mapping f takes the value b at the point a. Suppose X and Y are compact Riemann surfaces and f : X → Y is a non-constant holomorphic mapping. For x ∈ X, let v(x, f ) be the multiplicity with which f takes the value f (x) at the point x. The number b(f, x) := v(f, x) − 1 is called the branching order of f at the point x. In particular, b(f, x) is zero precisely if f is unbranched at x. Since X is compact, there are only finitely many points at which f is branched. Thus we can define the total branching order of f by X b(f ) := b(f, x) x∈X
Theorem(Riemann-Hurwitz): Suppose f : X → Y is an n-sheeted holomorphic covering mapping between compact Riemann surfaces X and Y with total branching order b. Let g be the genus of X and g 0 be the genus of Y , then we have the following formula b (6.6) g = + n(g 0 − 1) + 1 2 Proof: From a corollary of Riemann-Roch theorem, we have seen that the degree of canonical class on a compact Riemann surface of genus g is 2g − 2. Let ω be a nonvanishing meromorphic 1-form on Y . We have deg(ω) = 2g 0 −2 and deg(f ∗ ω) = 2g−2. Suppose x ∈ X and y = f (x) ∈ Y , by the previous theorem, we can find a coordinate neighborhood (U, z) of x and (U 0 , w) of y with z(x) = 0 and w(y) = 0 such that with respect to these coordinates f can be written as w = z k , where k = v(f, x). On U 0 let ω = ψ(w)dw, then on U we have, f ∗ ω = ψ(z k )dz k = kz k−1 ψ(z k )dz Hence, ordx (f ∗ ω) = (k − 1) + k · ordy (ω) = b(f, x) + v(f, x) · ordy (ω) Since X
v(f, x) = n
x∈f −1 (y)
for any y ∈ Y , we get X x∈f −1 (y)
ordx (f ∗ ω) =
X x∈f −1 (y)
58
b(f, x) + n · ordy (ω)
Thus, deg(f ∗ ω) =
X
ordx (f ∗ ω) =
X
ordx (f ∗ ω)
y∈Y x∈f −1 (y)
x∈X
=
X X
b(f, x) + n ·
x∈X
X
ordy (ω) = b(f ) + n · deg(ω)
y∈Y
This implies 2g − 2 = b + n · (2g 0 − 2), and the Riemann-Hurwitz formula follows. Now we apply the formula to the case of an n-sheeted covering from a compact ˆ with total branching order Riemann surface X of genus g over the Riemann sphere C b, it follows, b g = −n+1 2 ˆ then g = (b/2)−1 and for any branched point x ∈ X, If there is a double covering of C, its branching order b(f, x) is 1, hence b is exactly the number of branched points on ˆ is X. A compact Riemann surface of genus g > 1 admitting a double covering of C called hyperelliptic. This definition of hyperelliptic Riemann surface is equivalent to that given in first chapter, and we will prove the equivalence in next chapter. For example, for a polynomial P (z) = (z − a1 ) · · · (z − ak ) of degree k with distinct p roots, P (z) gives a Riemann surface X of hyperelliptic curve C. We have an explicit basis ω1 , · · · , ωg for H 0 (X, Ω) as z j−1 dz 1≤j≤g ωj := p P (z) In particular, if the degree of P (z) is 3 or 4, namely, C is an elliptic curve, then X has genus one and dz ω=p P (z) is a basis of H 0 (X, Ω). Let Λ be the corresponding lattice, the mapping J : X → Jac(X) = C/Λ is then given the elliptic integral of the first kind Z J(x) = a
x
dz p mod Λ ∈ Jac(X) P (z)
ˆ be Since J is an isomorphism, let π1 be the quotient map C → C/Λ and π2 : X → C the covering map, then ˆ f := π2 ◦ F ◦ π2−1 : C → C is a doubly-periodic meromorphic function. This fact connects elliptic integral to doubly-periodic functions, which we shall discuss later in next section. 59
6.3
Doubly Periodic Functions
In this section, we consider complex torus C/Λ where Λ is the lattice in C, which is the model for a compact Riemann surface of genus one. In next section, we will go through compact Riemann surface of genus zero (the Riemann sphere) and of genera at least two (hyperelliptic Riemann surfaces). The method we used in this section basically relies an adequate supply of doubly periodic functions, namely elliptic functions, which will not be extended to the situation when the torus is of higher dimension. The generalization will be introduced in Chapter 9. Let λ1 , λ2 be a pair of generators of Λ, with Im(λ1 /λ2 ) > 0, and let z0 ne a point of C; the interior of the parallelogram whose vertices are z0 , z0 + λ1 , z0 + λ2 and z0 + λ1 + λ2 is denoted by Γ. The point z0 is so chosen that none of the functions concerned with has poles or zeros on its boundary ∂Γ. A doubly periodic function with respect to λ is a meromorphic function f (z) on C such that f (z + λ) = f (z) for all λ ∈ Λ. In other words, f (z + λ1 ) = f (z + λ2 ) = f (z). It is a fundamental property of doubly periodic function that counting multiplicity the number of poles and that of zeros in Γ coincide. Moreover, suppose the poles and zeros are P1 , · · · , Pm and Q1 , · · · , Qn respectively, we have P1 + · · · Pm ≡ Q1 + · · · + Qn mod Λ To see this, we take the following integrals, Z Z X X zf 0 (z) f 0 (z) dz = 2πi(n − m), dz = 2πi( Qi − Pj ) fz ∂Γ ∂Γ f z Since f 0 (z)/f (z) is doubly periodic, we have z0 +λ2
Z
z0
Z
f 0 (z) dz + f (z)
z0 +λ2
{
= z0
z0 +λ1
Z
z0 +λ1 +λ2
f 0 (z) dz f (z)
f 0 (z) f 0 (z + λ1 ) − }dz = 0 f (z) f (z + λ1 )
Similarly, we have, z0 +λ1 +λ2
Z
z0 +λ2
Together, we have
R
f 0 (z) dz ∂Γ f z
f 0 (z) dz + f (z)
Z
z0
z0 +λ1
f 0 (z) dz = 0 f (z)
= 2πi(n − m) = 0, i.e. n = m.
Similarly, Z
z0 +λ2
z0
zf 0 (z) dz + f (z)
Z
60
z0 +λ1
z0 +λ1 +λ2
zf 0 (z) dz f (z)
(6.7)
Z
z0 +λ2
= z0
−λ1 f 0 (z) dz = −λ1 logf (z)|zz00 +λ2 = 2πi · n1 λ1 f (z)
where n1 ∈ Z. And, Z z0 +λ1 +λ2 z0 +λ2
zf 0 (z) dz + f (z)
Z
z0
z0 +λ1
zf 0 (z) dz = 2πi · n2 λ2 f (z)
Together, we obtain X
Qi −
X
Pj = n1 λ1 + n2 λ2 ≡ 0 mod Λ
Clearly, we cannot have the case that m = n = 1 for otherwise the only pole and the only zero would cancel each other. So the simplest non-constant doubly periodic function will have a double pole in Γ. In the following, we will construct such a function. Suppose f (z) is such a function, then f (z) − f (−z) is doubly periodic with at most a simple pole, hence must be constant; since it is odd, the only possibility is a zero function, i.e. f (z) is even. Consider transformation in the form a · f (z) + b where, a,b ∈ C, we can write f (z) = z −2 + O(z 2 ) near the origin. It is almost clear that by summing f (z) in the local expression near z = 0 over all z + λ where λ ∈ Λ, we will obtain the global expression of the desired function. However, to make sure that the sum will converge, before our construction, we first consider its derivative f 0 (z) = −2z −3 + O(z) near the origin. P We take the formal sum λ∈Λ 2(z +λ)−3 which is convergent to a function denoted by P 0 (z), where P(z) is the doubly periodic function we want to construct. Clearly P 0 (z) is odd and doubly periodic. Since P(z) has residue zero at every pole, its integral is single-valued, X 1 1 − } (6.8) { P(z) = z −2 + 2 2 (z + λ) λ λ∈Λ,λ6=0 Now P(z) is well-defined and is even. To see that is is doubly periodic, set g(z) = P(z + λ1 ) − P(z). Its derivative is zero hence the function is constant. Since P(z) is even, we have that g(− λ21 ) = 0; thus g(z) ≡ 0. This implies P(z + λ1 ) = P(z). Similarly, P(z + λ2 ) = P(z), i.e. P(z) is doubly periodic. This simplest doubly periodic function is called Weierstrass P-function. Now the function P 02 − P 3 is an even doubly periodic function with at most poles of order 2 at the points of Λ. Subtracting a suitable multiple of P, say g2 · P, we will obtain a holomorphic doubly periodic function, say h(z). h(z) − h(0) is also a doubly periodic function. If it is non-constant, h(z) − h(0) has no zeros nor poles, which is a contradiction since h(0) − h(0) = 0. Therefore, h(z) is a constant, say g3 . Now we have the following equation, P 02 = 4P 3 − g2 · P − g3 61
(6.9)
where constants g2 and g3 depend on Λ. Since P has residue 0 at every pole, it has a single-valued integral −ζ, where ζ is the Weierstrass zeta-function, Z z X 1 1 z 1 1 −1 ζ(z) = z − {P(u) − 2 }du = + − + 2} { (6.10) u z λ∈Λ,λ6=0 z + λ λ λ 0 We note ζ(z + λ) − ζ(z) is a constant for any λ ∈ Λ since its derivative vanishes. We define two constants η1 , η2 , ηi = ζ(z + λi ) − ζ(z) i = 1, 2 By induction, ζ(z + n1 λ1 + n2 λ2 ) − ζ(z) = n1 η1 + n2 η2
∀n1 , n2 ∈ Z
Taking integral we have the following Legendre’s relation, Z ζ(z)dz = 2πi = λ1 η2 − λ2 η1
(6.11)
∂Γ
Since the only singularities of ζ(z) are simple poles of residue 1 at points of Λ, its integral has logarithmic branching at each of these points. By taking exponential, we have obtain a single-valued function, Z z Y z+λ z z2 1 { exp(− + 2 )} (6.12) σ(z) = exp{logz + (ζ(u) − )du} = z u λ λ 2λ 0 λ∈Λ,λ6=0 which is called Weierstrass sigma function. It is odd, holomorphic and has only simple zeros at points of Λ. By integrating ζ(z + λi ) − ζ(z) = ηi on both sides and taking the exponential, we have σ(z + λi ) = Ci · σ(z) · exp(zηi ) i = 1, 2 where Ci are some constants. Setting z = − λ2i we have Ci = exp(− λi2ηi )−1 , therefore, σ(z + λi ) = −σ(z) · exp{ηi (z +
λi )} 2
It can be generalized to λ σ(z + λ) = (−1)n σ(z) · exp{η(z + )} 2 where λ = n1 λ1 + n2 λ2 , η = n1 η2 + n2 η1 and n1 , n2 are integers. With sigma function, we are going to prove the following theorem, which is a converse to the fundamental property of doubly periodic functions we just discussed. 62
Theorem: Let P1 , · · · , Pn and Q1 , · · · , Qn be points of Γ (not necessarily all distinct but no Qi and Pj coincide. Suppose we have, P1 + · · · + Pn ≡ Q1 + · · · + Qn mod Λ then there exists a doubly periodic function f (z) whose poles in Γ are just Pi and whose zeros are just Qi . This function can be written explicitly as f (z) =
σ(z − Q1 ) · · · σ(z − Qn ) σ(z − R1 ) · · · σ(z − Rn )
(6.13)
P P where R1 , · · · , Rn are so chosen that Ri ≡ Pi mod Λ and Ri = Qi . This theorem, together with its converse, is in some sense just the restatement of Abel’s Theorem for the case g = 1. Now we generalize this function to the family of theta functions, denoted by θ(z). By its definition, theta functions are holomorphic functions satisfying θ(z + λ) = θ(z)exp · F (z, λ) for any λ ∈ Λ, where F (z, λ) is an inhomogeneous function linear in z. Suppose θ(z) is not identically zero, and let Q1 , · · · , Qn be the zeros in Γ counting multiplicity, the function, ψ(z) =
θ(z) σ(z − Q1 ) · · · σ(z − Qn )
is a trivial theta function, namely, a theta function with no zeros. We finish this section with a fundamental theorem about trivial theta functions. Theorem(Trivial theta function): Any trivial theta function has the form eQ(z) for some quadratic polynomial Q(z) and conversely any such eQ(z) is a trivial theta function. Proof: Let θ(z) be a trivial theta function. Since it has no zero, any branch of logθ(z) is holomorphic satisfying logθ(z + λi ) − logθ(z) = F (z, λ1 ) = O(1 + |z|)
(6.14)
for the two generators λi , i = 1, 2. Since logθ(z) is bounded on Γ and it takes only O(1 + |z|) steps each ±λ1 or ±λ2 to go from any z to a point of Γ, it follows logθ(z) = O(1+|z|2 ). This implies that logθ(z) is quadratic polynomial. The converse is trivial. Consequently, any theta functions with prescribed zeros can be expressed in terms of sigma functions. 63
6.4
Poincar´ e Series
A compact Riemann surface of genus zero is isomorphic to a Riemann sphere, or equivalently, the projective 1-space P1 , and all meromorphic functions on P1 are rational functions, namely are those written as a quotient of two polynomials with non-zero denominator. First, we note polynomials are meromorphic functions on P1 . For a polynomial P (z) = cn z n +· · ·+c1 z+c0 , let w = z1 , then P ( w1 ) = wcnn +· · ·+ cw1 +c0 , i.e. P is holomorphic in a neighborhood of w = 0 with a pole of order n at ∞ on P1 . Now let f be a non-constant meromorphic function on P1 whose poles on C are distinct and isolated, denoted by P1 , · · · , Pm with order k1 , · · · , km . Let Q(z) = (z − P1 )k1 · · · (z − Pm )km , then Q(z) · f (z) has removable singularities at P1 , · · · , Pm . Then Q(z)f (z) is holomorphic on C with a pole of order l at ∞. Therefore |Q · f (z)/z l | is bounded at z = ∞, which means Q(z) · f (z) must be a polynomial of degree no larger than l. This show the meromorphic function f must be rational. Now for higher genera compact Riemann surfaces, we can embed them into complex manifolds Pn as submanifolds called projective varieties by meromorphic functions. For g ≥ 2, this is done with Poincar´e series, which will be introduced in this section. To construct non-trivial meromorphic function on ∆/Γ, suppose f = s/t is such a function written as a quotient of two holomorphic functions s and t on ∆. For f to be Γ-invariant over ∆, we will need some functional equation for s and t. ∀ γ ∈ Γ, we denote γ(z) by γz for any z ∈ ∆. We write the following functional equation for a holomorphic function s on ∆, s(γz ) = αγ (z)s(z)
(6.15)
where αγ (z) 6= 0 satisfying the following condition: For any two elements γ and µ in Γ, we have, by the functional equation that s(γ · µ(z)) = αγµ (z)s(z), while, on the other hand, by applying functional equation separately we have s(γ · µ(z)) = s(γ(µ(z))) = αγ (µz )s(µz ) = αγ (µz )αµ (z)s(z). Hence we have αγµ (z) = αγ (µz )αµ (z) (6.16) If α : Γ × ∆ → ∆ satisfies this equation for any γ,µ ∈ Γ them {αγ (z)}γ∈Γ is called a system of factors of automorphy. Now we consider the holomorphic p-differential s(z)dz p , which is invariant under Γ, namely, s(γz )(dγz )p = s(z)(dz)p this implies s(γz )(γ 0 (z))p (dz)p = s(z)(dz)p , thus s(γz )(γ 0 (z))p = s(z). By setting 1 0 0 0 αγ (z) = (γ 0 (z)) p , (6.15) is satisfied. Moreover, by chain rule, (γ · µ(z)) = γ (µz ) · µ (z), 64
hence (6.16) also holds, i.e. the {αγ }γ∈Γ so defined forms a system of factors of automorphy. Now it is easy to check that our f (z) = s(z)/t(z) with s and t satisfying the functional equation (6.15) is invariant under Γ: s(γz ) s(z)αγ (z) = = f (z) t(γz ) t(z)αγ (z)
f (γz ) =
Definition(Poincar´e series): Given a holomorphic function f on ∆, for k ∈ N+ , the Poincar´e series is defined as X PΓk (f )(z) = f (γz )(γ 0 (z))k (6.17) γ∈Γ
Lemma: For any k ≥ 2, the Poincar´e series converges on ∆. P absolutely 0 k Proof: Since f is bounded, it suffices to show that Γ |γ (z)| converges for k ≥ 2. Recall that for a holomorphic function ϕ, it can be regarded as a real map in R2 such that det(dϕ) = |ϕ0 |2 , hence when k = 2, |γ 0 (z)|2 represents a change of area, therefore, we have, for some bounded domain D = ∆(z, ) Z X X Area of γ(D) |γ 0 (z)|2 = D
γ
γ
P P Since ∆/Γ is compact, the change of area γ |γ 0 (z)|2 = γ Area of γ(D)/Area of γ(D) is bounded. For k ≥ 2, the situation is even better. So we conclude that for k ≥ 2, the Poincar´e series is absolutely convergent. Now we denote a convergent Poincar´e series by s(z), we check that it satisfies the functional equation of (6.15). For some µ ∈ Γ, we have X s(µz ) = f (γµ(z) )(γ 0 (µz ))k γ∈Γ
By chain rule, we have (γµ)0 (z) k ) f (γµ(z) )(γ (µz )) = f (γµ(z) )( 0 µ (z) γ∈Γ γ∈Γ
X
0
=
k
X
X
f (δz )(
δ∈Γ
= where δ = γµ, i.e. s(µz ) =
δ 0 (z) k ) µ0 (z)
X 1 f (δz )(δ 0 (z))k 0 k (µ (z)) δ∈Γ
1 s(z). (µ0 (z))k
65
If we can find some non-zero convergent Poincar´e series, then by taking quotient of two such series we will obtain a meromorphic function which is invariant under Γ. For a fixed base point z0 ∈ ∆, let Γ0 be the subset of Γ consisting of all γ0 such that |γ00 (z0 )| > 12 . Suppose there is a polynomial f such that f (z) = 1 when z = z0 and f (γz0 ) = 0 for any γ ∈ Γ0 . Then X X s(z0 ) = f (γz0 )(γ 0 (z0 ))k + f (γz0 )(γ 0 (z0 ))k γ∈Γ0
γ∈Γ−Γ0
=1+
X
f (γz0 (γ 0 (z0 ))k )
id6=γ∈Γ−Γ0
By convergence of Poincar´e series for k = 2, we have M ≥ 0. Then we have X id6=γ∈Γ−Γ0
|f (γz0 )| · |γ 0 (z0 )|k =
X id6=γ∈Γ−Γ0
P
|f (γ0 )| · |γz0 0 |2 ≤ M for some
1 |f (γz0 )| · |γ 0 (z0 )|2 · |γ 0 (z0 )|k−2 ≤ ( )k−2 · M 2
Hence, for k large enough, |s(z0 )| ≥ 1 − ( 12 )k−2 · M >= 0. Similarly, we can construct some t(z) which does not vanish at z0 , then the function f (z) = st (z) is a non-trivial meromorphic function. k k ) the cotangent line bundle of X to the power of k and Γ(X, KX Denote by KX the corresponding sections, then we see that for a convergent Poincar´e series S(z), k ). S(z)(dz)k descents to some σ ∈ Γ(X, KX
Now we proceed to embed X = ∆/Γ into PN for some N by means of sections σi ∈ k Γ(X, KX ) for 0 ≤ i ≤ N . Recall that for a line bundle L over a compact manifold X with a covering {Uα }α , we have L|Uα ∼ = Uα ×C. Let eα be a basis for Uα corresponding to (x, 1) ∈ Uα × C. Suppose σ0 , σ1 , · · · , σN ∈ Γ(X, L), and σi |Uα = Si,α eα , where Si,α are holomorphic functions on Uα , then Si,α (x) = ϕαβ (x) · Si,β (x) for any x ∈ Uα ∩ Uβ = Uαβ . Consider the map Φ : X → PN such that x 7→ [σ0 (x), · · · , σN (x)] for any x ∈ Uα ⊂ X. The map is well-defined provided Si,α (x) 6= 0 for at least one index, for we have (S0,α (x), · · · , SN,α (x)) = ϕαβ (x) · (S0,β (x), · · · , SN,β (x)) for any x ∈ Uαβ , k thus [S0,α (x), · · · , SN,α (x)] = [S0,β (x), · · · , SN,β (x)] ∈ PN . Now take L to be KX for some power k large enough, we claim that Φ is a holomorphic embedding, namely, it is an injective immersion. To prove the injectivity of Φ, let x,y ∈ X = ∆/Γ be two distinct points with x = π(z1 ) and y = π(z2 ), where π : ∆ → X is the projection. It suffices to find two sections k σ1 = S1 (z)(dz)k , σ2 = S2 (z)(dz)k ∈ Γ(X, KX ), where Si are Poincar´e series, such that Si (zj ) = δij . To do so, we take a subset Γ0 := {γ ∈ Γ : |γ 0 (z1 )| > 21 or |γ 0 (z2 )| > 12 } of Γ. For S1 , we take f1 to be a polynomial such that f1 (z1 ) = 1, f1 (γz1 ) = 0 for any id 6= γ ∈ Γ0 and f1 (γz2 ) = 0 for any γ ∈ Γ0 (including id), which can be done through Lagrange interpolation, and is possible since there is no γ such that γz2 = z1 . 66
Similarly, we can take f2 for S2 to be a polynomial such that f2 (z2 ) = 1, f2 (γz2 ) = 0 for any id 6= γ ∈ Γ0 and f2 (γz1 ) = 0 for any γ ∈ Γ0 . Now given some small > 0, for k large enough, we have |S1 (z1 ) − 1| < and |S2 (z2 ) − 1| < while |S1 (z2 )| and |S2 (z1 )| are both smaller than . This shows images of x and y under Φ are separated. To show it is an immersion, we take some x0 ∈ X. For k large enough, ∃ σ0 ∈ k Γ(X, KX ) arising form S0 (z) = PΓk (f )(z) such that σ0 (x0 ) 6= 0, i.e. S(z0 ) 6= 0 for any k z0 ∈ ∆ projected onto x0 . We want to produce a σ1 ∈ Γ(X, KX ) such that σσ01 (x0 ) 6= 0. This will be done by constructing a Poincar´e series for σ1 sufficiently small at x0 whose derivative does not vanish. For this purpose, we can find a holomorphic function h on ∆ such that h(z0 ) = 0 and h0 (z0 ) = 1, while h(γz0 ) = h0 (γz0 ) = 0, for any γ ∈ Γ0 −{id}, where, as usual, Γ0 denotes the subset of Γ whenever |γ 0 (z0 )| > 12 (including id). Now S1 (z) = PΓk (h) can be decomposed into two parts, summing over γ in Γ0 (denoted by S10 ) and out of Γ0 (denoted by S100 ) respectively. Clearly S10 (z0 ) = 0 while S100 (z0 ) is d 0 d 00 small when k is large enough. On the other hand dz S1 (z0 ) = h0 (z0 ) = 1 while dz S1 (z) is small in a neighborhood of z0 by Cauchy estimation. This completes our proof that Φ is an embedding from X into PN .
6.5
Projective Embedding of Compact Riemann Surfaces
With Serre duality theorem and Riemann-Roch theorem, we are able to prove an important theorem about compact Riemann surfaces, namely, all compact Riemann surfaces can be embedded into the projective space PN . We give a classical proof in this section, and a more modern proof will be given in the last chapter, and the statement turns out to be a special case of Kodaira embedding theorem prescribing the condition when a compact complex manifold is embeddable. Let X be a compact complex manifold of dimension n and let L be a holomorphic line bundle over X. As usual, we denote by Γ(X, L) the space of holomorphic sections of L. Then Γ(X, L) is a finite dimensional space, say of dimension N + 1, and we can have a map into PN (Γ(X, L)∗ ). We note, by choosing a basis of Γ(X, L)∗ , we can identify Γ(X, L)∗ with CN +1 , thus we can also identify P(Γ(X, L)∗ ) with PN . This gives P(Γ(X, L)∗ ) the structure of a complex manifold isomorphic to PN . For P ∈ X, define ΓP (X, L) be the space of holomorphic sections of L vanishing at P , which is a subspace of Γ(X, L). ΓP (X, L) has dimension N unless all sections of L vanish at P , in which case it has dimension N + 1. Consider the annihilator of ΓP (X, L) under the pairing Γ(X, L) × Γ(X, L)∗ → C. This subspace of Γ(X, L)∗ is a line if ΓP (X, L) is of dimension N or a zero if ΓP (X, L) is of dimension N + 1. In the first case, it is an element of P(Γ(X, L)∗ ). 67
A point P ∈ X becomes a base point of L if all holomorphic sections of L vanish at P and the set of base points of L is called the base locus denoted by B(L). If B(L) = ∅, L is base point free. We now define a map φ : X − B(L) → P(Γ(X, L)∗ ). Assume P ∈ X is not a base point, and choose a local frame χ for L on an open neighborhood U of P . Define a functional f on sections of L which acts on a section s by restricting s to U , writing s = x · σ on U for some holomorphic map σ : U → C, and evaluating σ at P . In other words χ(S) = σ(P ). The functional f annihilates ΓP (X, L), and is not identically zero since P is not a base point. Thus χ generates the one-dimensional subspace φ(X) ⊂ Γ(X, L)∗ . To write down the homogeneous coordinates of f with respect to the identification P(Γ(X, L)∗ ) ∼ = PN given by the basis s0 , · · · , sN for Γ(X, L), we simply evaluate χ at each si . Write si locally as ξ · σi , then χ(si ) = σi (P ). Thus, φ(P ) has homogeneous coordinates [σ0 (P ), · · · , σN (P )]. Hence our map φ : X − B(L) → P(Γ(X, L)∗ ) ∼ = PN is given by φ(P ) = [s0 (P ), · · · , sN (P )]. The local expression for φ makes it clear that the map is holomorphic since each of its coordinates si is given locally by a holomorphic function σi of P . Theorem: Let X be a compact Riemann surface, there exists an embedding of X into projective space PN . Let D be a divisor of X we start with a basis f0 , · · · , fN for H 0 (X, OD )). For P ∈ X, we choose a coordinate z with z(P ) = 0 and let k be the minimal order of {fi } at P , then we can write fi = z k · gi with gi holomorphic at P . Define: F (P ) := [g0 (P ), · · · , gN (P )] This definition is independent of the choice of z. We note P is not a base point if and only if k = −D(P ), but even if P is a base point, the map still makes sense. If P ∈ X is not a base point, we want to show that φ(P ) = (P ). Let s0 , · · · , sN be a basis for Γ(X, L[D]). To obtain the corresponding basis f0 , · · · , fN for H 0 (X, OD ), we choose a covering {Uα } of X and functions hα on each Uα with (hα ) = D on each Uα . We take our first Uα to be a coordinate neighborhood U of P such that intersection of D and U only contains P and let h = z nP on U . Extend to a covering {Uα } and functions {hα } with the required properties; this gives us a basis f0 , · · · , fN for H 0 (X, OD ) corresponding to s0 , · · · , sN . On U we have fi (z) = z −nP · si (z). Thus, φ(P ) = [s0 (P ), · · · , sN (P )] = [g0 (P ), · · · , gN (P )] = F (P ) where, as above, fi (z) = z −nP · gi (z). Now we prove a lemma, and our main theorem directly follows. 68
Lemma: Let D be a divisor of degree at least 2g + 1. If f0 , · · · , fN is a basis for H 0 (X, OD ), the associated map F into PN is an embedding. Proof: First, recall that for a meromorphic 1-form ω on X of genus g, its divisor D is linear equivalent to the canonical divisor C, and as a consequence of Riemann-Roch theorem, we have see deg(C) = 2g−2. Moreover, for a divisor D with deg(D) > 2g−2, we have H 1 (X, OD ) = 0. Indeed, as a result of Riemann-Roch theorem, we have seen that not all holomorphic differentials vanish at a point simultaneously on a compact Riemann surface, thus we can choose a non-trivial holomorphic 1-form ω, whose divisor is the canonical divisor C. By multiplication by ω, we have a sheaf isomorphism between OC−D and Ω−D . Since deg(C) = 2g − 2, and thus deg(C − D) < 0, we conclude that OC−D has no global sections, whence H 0 (X, OC−D ) = H 0 (X, Ω−D ) = 0. By Serre duality, we also have H 1 (X, OD ) = 0. To prove the injectivity of F , let P1 6= P2 be two distinct points of X. Define a divisor D0 by subtracting the point P2 form D, the degree of D0 is still at least 2g. So we have H 1 (X, O)) = 0. Thus, by Riemann-Roch theorem, the sheaf OD0 is globally generated in the sense that for each P ∈ X, these exists a global f ∈ H 0 (X, OD0 ) with the degree of f at P equal to −D0 (P ). To see this, suppose there is no f ∈ H 0 (X, OD0 ) with ordP (f ) = −D0 (P ), then we would have H 0 (X, OD0 ) = H 0 (X, OD0 −P ). But since deg(D0 ) and deg(D0 −P ) are both larger than 2g−2, the first cohomology groups of OD0 and OD0 −P both vanish. Riemann-Roch theorem implies dimH 0 (X, OD0 ) = 1 − g + deg(D0 ) 6= 1 − g + deg(D0 − P ) = dimH 0 (X, OD0 −P ) which is a contradiction. 0 In particular, there exists such an f for P = P1 . Now, P since f ∈ H (X, OD ) as well, f can be expressed as a linear combination f = λi · fi . Since {fi } are a basis 0 for H (X, OD ) and since this sheaf is globally generated, at least one fi has minimal k (j) order kj = nPj for j = 1, 2. As in the definition of F , write fi = zj j · gi , where zj is a k coordinate with zj (Pj ) = 0. Similarly, write f = zj j ·g (j) and we note g (1) (P1 ) 6= 0. On the other hand, since f ∈ H 0 (X, OD0 ), we have ordP2 (f ) ≥ −D0 (P2 ) = −k2 −1. Thus, P (j) (j) (j) g2 (P2 ) = 0. Now, g (j) (P ) = λi · · · gi (Pj ), and F (Pj ) = [g1 (Pj ), · · · , gN (Pj )]. j P Write y(t1 , · · · , tN ) = λi · ti ; y is a homogeneous polynomial of degree 1. Then (j) g (Pj ) = 0 if and only y(F (Pj )) = 0. Hence y(F (P1 )) 6= 0 and y(F (P2 )) = 0, and thus F (P1 ) and F (P2 ) cannot be the same point in projective space.
To prove that F is an immersion, we consider a point P0 ∈ X. Construct the divisor D0 as above by subtracting P0 from D. Since OD0 is globallyPgenerated, we get f as above with ordP0 (f ) = −D0 (P0 ) = −D(P0 ) + 1. We have f = λi · fi . Write fi = z −nP0 · gi −nP0 where z is a coordinate with z(P · g near P0 ; g has a P0 ) = 0, and also write f = z zero of order 1 at P0 , and g = λi · gi . As OD0 is globally generated, at least one gi is non-zero at P0 ; without loss of generality, assume that g0 (P0 ) 6= 0. Then, using one 69
of the standard charts on PN , we can write F = (F1 , · · · , FN ) = ( gg01 , · · · , ggN0 ) near P P0 . Thus, in this representation, we have λi · Fi = gg0 − λ0 . Taking the differential P of this expression, we see that λi · · · dFi = d gg0 , which is non-vanishing since g vanishes to the order of 1 at P0 and g0 (P0 ) 6= 0. Thus, not all dFi are zero, and we conclude dF is not the zero linear transformation at P0 . This completes our proof to the lemma, and the main theorem follows directly.
70
Chapter 7 Abel Map and Jacobi Variety 7.1
The Jacobian and Abel’s Theorem
Definition(Lattice): Suppose V is an n-dimensional vector space over R. An additive subgroup Λ is called a lattice if there exist n vectors v1 , · · · , vn ∈ V , which are linearly independent over R such that Λ = Zv1 + · · · + Zvn . Let X be a compact Riemann surface of genus g, recall the 4g-gon model we used before, the corresponding basis of H1 (X, Z) is {ai ,Rbi } for i = 1, 2, · · · , g. Let ω1 , · · · , ωg be a normalized basis of H 0 (X, Ω) satisfying aj ωi = δjk . Let Λ ne the subgroup R R of Cg consisting of vectors λγ = ( γ ω1 , · · · , γ ωg ) where γ runs over H1 (X, Z). We have λak = (0, · · · , 1, · ·R· , 0) = ek ,Rthe vector in Cg with 1 in the k-th place and 0 elsewhere, and λbk = ( bk ω1 , · · · , bk ωg ) = Bk consists of the columns of the maR trix B = (Bjk ), where Bjk = bj ωk . Since Im(B) is positive definite, the 2g vectors {e1 , · · · , eg , B1 , · · · , Bg } are linearly independent over R. Moreover, since {ai , bi } generate H1 (X, Z), it follows Λ = Ze1 + · · · + Zeg + ZB1 + · · · + ZBg . This implies that Λ is a lattice in Cg with a compact quotient Jac(X) = Cg /Λ. P The divisor of an abelian differential ω is (ω) = P ∈X ord(P ) · P . Since the quotient of two abelian differentials is a meromorphic function, any two divisors of abelian differentials are linearly equivalent, which correspond to a canonical class denoted by C. Fix a base PNpointRPP0j ∈ X, the Abel map is defined for divisors in a natural way, A(D) = j=1 nj P0 ω ~ . If the divisor D is of degree zero, then A(D) is independent of P0 , for we can write D into D = P1 + · · · + PN − Q1 − · · · − QN (allowing possible P R Pi multiplicities) so that A(D) = N ~ . The following theorem, which is known i=1 Qi ω as Abel’s theorem is central in study of the relationship between a compact Riemann 71
surface X and its Jacobian Jac(X). Theorem(Abel): Let D be a divisor on a compact Riemann surface X of degree zero, then D is linearly equivalent to zero if and only if A(D) = 0. P PN Since the degree is zero, we write D = N i=1 Pi − i=1 Qi , and the theorem says the necessary and sufficient condition that there exists a meromorphic function f with P P Pi as its divisor of zeros and Q as its divisor of poles (i.e. (f ) = P1 + · · · + PN R Pii PN − Q1 − · · · − QN ) is that i=1 Qi ω ~ ≡ 0 mod Λ. proof: To show the necessity, let D = (f ) for some meromorphic function f on X. Since poles with residue +1 and −1 can be paired in this case,R it follows that the abelian differential µ := dff = d(log(f )) is of third kind. Clearly, γ dff ∈ 2πiZ, for any R R closed curve γ on X, hence Ak (µ) = ak µ = 2πi · nk and Bk (µ) = bk µ = 2πi · mk with mk , nk ∈ Z. RP From the proof of Reciprocity theorem (1) with uj (P ) = P0 ωj , we have, Z uj · µ = 2πi
X
∂∆
(uj (Pk ) − uj (Qk )) = 2πi
XZ k
k
Pk
ωj
Qk
On the other hand, from (5.7) with ϕ = µ we have, Z uj · µ = ∂∆
g X
{Ai (ωj )Bi (µ) − Bi (ωj )Ai (µ)}
i=1
= Bj (µ) −
g X
Bi (ωj )Ai (µ)
i=1
= 2πi · nj −
g X
Z 2πi · mi
ωj bi
i=1
Together,
PN R Pk k=1
Qk
ωj = nj −
Pg
i=1
mi
R bi
A(D) =
ωj , which gives the following result,
N Z X k=1
N Z X =( k=1
Pk
ω1 , · · · ,
Qk
Pk
ω ~
Qk N Z X k=1
Pk
ωg )
Qk
= (n1 · e1 + · · · ng · eg ) − (m1 · B1 + · · · + mg · Bg ) ≡ 0 mod Λ To prove the sufficiency, let D = P1 + · · · + PN + Q1 + · · · + QN , and assume A(D) = PN R Pi i=1 Qi ω ≡ 0 mod Λ. We choose curves joining Pk and Qk which does not intersect ai , bi , i = 1, · · · , g. Consider the normalized abelian differential of third kind µPk Qk 72
and define µ = are,
PN
k=1
µPk Qk . Clearly, µ has all zero a− periods and its b− periods N Z N Z X X ( µPk Qk , · · · , µPk Qk ) k=1
b1
k=1
=
N Z X k=1
N Z X =( k=1
bg
Pk
ω ~
Qk
Pk
ω1 , · · · ,
Qk
N Z X k=1
Pk
ωg )
Qk
= (n1 · e1 + · · · ng · eg ) + (m1 · B1 + · · · + mg · Bg ) Pg PN ω · m = where m ,n ∈ Z, i = 1, · · · , g. Now, define η = µ − i i i i i=1 k=1 µPk Qk − Pg ω · m , we have, i i=1 i Z X Z X Z Z N N µPk Qk )−m1 B1 −· · ·−mg Bg = (n1 , · · · , ng ) η) = ( µPk Qk , · · · , ( η, · · · , b1
bg
b1 k=1
bg k=1
Hence, all the b− R periods of η are integers. And since (ai , bi ) i = 1, · · · , g is the basis for H1 (X, Z), γ 2πiη ∈ 2πiZ for any closed curve γ on X. Then (f ) = D, where RP f (P ) = exp( 2πiη) is a meromorphic function on X. In fact, by linearity of Abel map, all linearly equivalent divisors are mapped by Abel map to the same point of the Jacobian Jac(X).
7.2
Jacobi Inversion Problem
Let Div(X) be the abelian group of divisors on X and Div0 (X) be the subgroup of all divisors on X with degree 0. We have seen that Abel map A : Div0 (X) → Jac(X) defines a group homomorphism whose kernel is precisely the subgroup of all the principal divisors, denoted by Divp (X). It induces an injective homomorphism j from the quotient group Div0 (X)/Divp (X) into Jac(X). This quotient group, denoted by Pic0 (X) is a subgroup of Picard group Pic(X) := Div(X)/Divp (X). We will see in this section that this map is indeed an isomorphism and we proceed to prove the surjectivity. From last section, there is no point on a compact Riemann surface such that all the holomorphic forms vanish. This enable us to pick up g points on a compact Riemann surface X of genus g such that no non-trivial holomorphic forms vanish at all the g points. To see this, we define a subspace as following, ∀a ∈ X, let Ha := {ω ∈ Ω(X) | ω(a) = 0} 73
Every H Ta has codimension 1 in Ω(X) since not all ω ∈ Ω(X) are in Ha . On the other hand, a∈X Ha = 0, and dim Ω(X) = g. We can select an arbitrary point a1 ∈ X so that dim Ha = g − 1; Suppose we have 1 ≤ k < g points a1 , · · · , ak ∈ X with dim (Ha1 ∩· · ·∩Hak ) ≤ g−k, since its intersection with all the remaining Ha goes to 0, we can find at least one ak+1 different from a1 , · · · , ak so that dim (Ha1 ∩· · ·∩Hak +1 ) ≤ dim (Ha1 ∩ · · · ∩ Hak ) − 1 ≤ g − k − 1. By induction, we can find g points so that Ha1 ∩ · · · ∩ Hag = 0. Now we take g disjoint simply connected coordinate neighborhoods (Uj , zj ) around each aj with zj (aj ) = 0. On each Uj , let ωi = ϕij dzj . The matrix A := (ϕij (aj ))1≤i,j≤g is invertible by our choice of a1 , · · · , ag . We define a map F from U1 × · · · × Ug into Cg as follows: ∀(x1P , · · · ,Rxg ) ∈ U1 × xj g · · · × Ug , F (x1 , · · · , xg ) = (F1 (x1 ), · · · , Fg (xg )) where Fi (xi ) := j=1 aj ωi . The Rx integral ajj ωi is along any curve γj from aj to xj lying in Uj . The Jacobian matrix of this map is the matrix A defined above, which is invertible at a = (a1 , · · · , ag ) with F (a) = 0 ∈ Cg . By inverse mapping theorem, F maps a neighborhood W of a ∈ U1 × · · · × Ug injectively onto a neighborhood F (W ) of 0 ∈ Cg . g Let p ∈ Jac(X) be an arbitrary point represented by the vector Pgξ ∈ C . For N ∈ Z large enough, the vector ξ/N lies in F (W ), i.e. for γ := j=1 γj we have R R ( γ ω1 , · · · , γ ωg ) = ξ/N . The divisor D := ∂γ lies in Div0 (X) and is mapped to ξ/N (mod Λ) under j. Let θ be the point of Pic0 (X) represented by N D, we have j(θ) = p ∈ Jac(X), which completes the proof.
Moreover, if g > 0, the Abel map A : X → Jac(X) is an embedding. Proof: If P, Q ∈ X and A(P ) = A(Q), then by the linearity of Abel map and Abel’s theorem, A(P − Q) = 0, hence there is a meromorphic function f on X with (f ) = P − Q, i.e. f only has a single simple pole. Therefore, as we have seen ˆ which has genus zero, before, this implies X is isomorphic to the complex R x sphere C, Rx a contradiction. Moreover, since A is given by ( P0 ω1 , · · · , P0 ωg ), let ωk = fk dz in a local coordinate, the tangent of A at P ∈ X is given by (f1 (P ), · · · , fg (P )). Since ω1 , · · · , ωg form a basis for the space of holomorphic 1-forms and in last section we have seen that there is no such a point on a compact Riemann surface at which all holomorphic 1-forms vanish, ωk ’s cannot all be zero at a same point, the same with fk ’s. Therefore dA is also injective, and A is an embedding. As a generalization, we now study the relation between Jac(X) and X g . Definition: Let Sn be the symmetric group on n letters. Sn acts on X n = X ×· · ·×X by permutation; the quotient S n (X) := X n /Sn is called the symmetric power of X, which is a complex manifold of dimension n. Now we proceed to introduce coordinates on S n (X) in the sense that points on X n 74
fixed by a non-trivial element of Sn shall have same coordinates. Consider the action of Sr on Cr and a neighborhood of the origin in Cr . Any germ of holomorphic function F at the origin invariant under Sr is a holomorphic function of the elementary symmetric functions in the coordinates z1 , · · · , zr of Cr , i.e., it is a holomorphic function of 1 2 1 (z1 + · · · + zr2 ), · · · , wr = (z1r + · · · + zrr ) 2! r! r And we take these w1 , · · · , wr as coordinates for C /Sr . For P1 , · · · , Pn ∈ X, we number them so that P1 = · · · = Pr1 = Q1 , Pr1 +1 = · · · = Pr1 +r2 = Q2 , · · · , Pr1 +···+rp−1 +1 = · · · = Pr1 +···+rp = Qp , with r1 +· · ·+rp = n, and Q1 , · · · Qp are distinct. The group of elements of Sn fixing (P1 , · · · , Pn ) is the sub-group Sr1 × · · · × Srp where Srk acts by permutation on the k-th block of rk points, so that a neighborhood of the image (P1 , · · · , Pn ) in S n (X) is isomorphic to a neighborhood of (0, · · · , 0) in Cr1 /Sr1 × · · · × Crp /Srp , ad we can use the coordinates mentioned above for each factor. w1 = z1 + · · · + zr , w2 =
P Consider the Abel map A : X g → Jac(X) by (P1 , · · · , Pg ) 7→ gk=1 A(Pk ). It induces a map A : S g (X) → Jac(X). We shall identify S g (X) with the set of effective divisors of degree g. Theorem: A : S g (X) → Jac(X) is a birational map; there is an analytic set Y ⊂ S g (X) of dimension less then g such that A : S g (X) − Y → Jac(X) − A(Y ) is an analytic isomorphism. To prove the theorem, we introduce some lemmas. Lemma 1: Let X be a Riemann surface, L be a holomorphic line bundle on X and V be a vector subspace of H 0 (X, L) of dimension k. Then there are k points P1 , · · · , Pk ∈ X such that if s ∈ V and s(Pi ) = 0 for i = 1, · · · , k, then s ≡ 0. Proof: If k > 0, let s1 ∈ V , s1 6≡ 0 and let P1 ∈ X be so that s1 (P1 ) 6= 0. Then V1 := {s ∈ V | s(P1 ) = 0} is mot all of V , so has dimension k − 1. If k − 1 > 0, choose s2 ∈ V1 and P2 ∈ X with s2 (P2 ) 6= 0. Then V := {s ∈ V1 | s(P2 ) = 0} = {s ∈ V | s(P1 ) = s(P2 ) = 0} has dimension k − 2. To obtain the conclusion, it suffices to iterate this process. P Lemma 2: The set of distinct points P1 , · · · , Pg ∈ X such that the rank at D = Pi of the differential of A : S g (X) → Jac(X) is maximal, equal to g, is open and dense in X g . Proof: Let (Uj , zj ) be coordinates near Pj with zj (Pj ) = 0. Using the local coordinates on Jac(X) coming from Cg , the map A can be written as X Z zj A(z1 , · · · , zg ) = ω ~ (7.1) j
where ω ~ = (ω1 , · · · , ω ~ = f~j dzj on Uj , with f~j = (fj1 , · · · , fjg ), the Jacobian Pg ). If ω matrix of A at D = Pi is given by 75
f~1 (P1 ) .. . f~g (Pg )
The existence of (P1 , · · · , Pg ) such that this matrix has rank g follows from Lemma 1, so does the fact that the set is dense. P Lemma 3: If D = gi=1 Pi ∈ S g (X), then A−1 A(D) is the bijective holomorphic image of Pr with r = dim|D| := H 0 (X, OD ) − 1. In particular, A−1 A(D) is connected for any D. Before giving the proof, we introduce some terminology. Let X be a compact Riemann surface and L be a holomorphic line bundle on X. For a non-trivial vector subspace V of H 0 (X, L), we call the set of effective divisors {D| D = Div(s) for some s ∈ V } the linear system determined by V . If V = H 0 (X, L), we call it the complete linear system of L. If L = L[D] for some divisor D, then this complete linear system consists of all effective divisors D0 ≥ 0 linearly equivalent to D. This is called the complete linear system of D and is denoted by |D|. And we write dim|D| = dimH 0 (X, OD ) − 1, and |D| is in one-one correspondence with the projective space (H 0 (X, L[D]) − {0}) /∼ := P(H 0 (X, L[D])). If dimH 0 (X, OD ) > 0 and by Serre duality, dimH 1 (X, OD ) = dimH 0 (X, Ω−D ) > 0, we call D a special divisor, which means that both D and KX − D, where KX is a canonical divisor on X, are linearly equivalent to effective divisors. Now we prove the lemma. If D1 , D2 ∈ S g (X) and A(D1 ) = A(D2 ), then D1 is linearly equivalent to D2 by Abel’s theorem and the linearity of Abel map, since in this case A(D1 − D2 ) = A(D1 ) − A(D2 ) = 0 and thus D1 − D2 is principal. Let Pr = P(H 0 (X, OD )) be the projective space (H 0 (X, OD )) − {0}) /∼, for some D ∈ S g (X). As we have seen, the map H 0 (X, OD ) − {0} → S g (X), s 7→ Div(s) induces a bijection between Pr and A−1 A(D). Indeed, this map is holomorphic. To see this, let U ⊂ CN be an open set, and f (x, t) be a function holomorphic on ∆ × U where ∆ is a disk of radius . Suppose f (x, t) 6= 0 for |x| = ρ < , t ∈ U . Then, if t0 ∈ U , the number of zeros xi (t) of f (x, t) counted with multiplicity in |x| < ρ is constant for t near t0 , say k, and for m ≥ 0, m ∈ Z, k X i
1 (xi (t)) = 2πi m
Z |x|=ρ
xm
∂x f (x, t) dx f (x, t)
so that this sum is holomorphic in t. Applying this to the zeros of a general section t0 s0 + · · · + tr sr ∈ H 0 (X, OD ) with respect to a basis (sj ) of H 0 (X, OD ), we get the desired result. 76
Now we prove the theorem. By Lemma 2, the set Y := {D ∈ S g (X)| rank(dA) at D < g} is an analytic set of dimension less than g. By Lemma 3, A−1 A(D) = {D} if D ∈ S g (X) − Y , it follows A restricted to S g (X) − Y is an isomorphism. We also note that if D ∈ S g (X) − Y , then dimH 0 (X, OD ) = 1. Indeed, Lemma 3 implies that if D is an isolated point in A−1 A(D), then dim|D| = 0. By Riemann-Roch theorem, we have dimH 0 (X, OD ) − dimH 0 (X, OC−D ) = 1 − g + deg(D) = 1 which means dimH 0 (X, OD ) = 1, and dimH 0 (X, OC−D ) = 0. Consequently, Y consists exactly of the special divisors of degree g. Theorem: For any D ∈ S g (X), the rank of the map A : S g (X) → Jac(X) at D equals g − dim|D|. P Proof: We take a divisor D of degree g, written as D = ni=1 ri Pi , with 0 < ri ∈ Z, P ri = g, P1 , · · · , Pn being distinct. We take a coordinates at D on S g (X) as we did before (using elementary symmetric functions), 1 r 1 (1) (1) (z + · · · + zrr11 ) = w1 = z1 + · · · + zr1 , w2 = (z12 + · · · + zr21 ), · · · , wr(1) 1 2! r1 ! 1 1 r2 1 (2) (2) w1 = zr1 +1 +· · ·+zr1 +r2 , w2 = (zr21 +1 +· · ·+zr21 +r2 ), · · · , wr(2) = (z +· · ·+zrr12+r2 ) 2 2! r2 ! r1 +1 ··· so on and so forth with z1 , · · · , zg being coordinates on X at P1 , · · · , P1 , · · · , Pn , · · · , Pn . | {z } | {z } r1
rn
Let ω1 , · · · , ωg a normalized basis of H 0 (X, Ω). If ωk = ff dz near P1 , then, for 1 ≤ i ≤ r1 , we have, by Taylor expansion Z zi Z zi z z (1) (2) (r ) {fk (P1 ) + fk (P1 )z + fk (P1 ) + · · · + fk 1 (P1 ) + · · · }dz ωk = constant + 2! r1 ! P0 = constant + zi fk (P1 ) + Hence, r1 Z X i=1
zi
zi2 (1) z r1 (r −1) fk (P1 ) + · · · + i fk 1 (P1 ) + · · · 2! r1 !
(1)
(r1 −1)
(1) (1)
ωk = constant + w1 fk (P1 ) + w2 fk (P1 ) + · · · + wr(1) fk 1
P0
It then follows, g Z X i=1
zi
ωk = constant +
P0
n X X
(ν) (j−1)
wj fk
+ O(w2 )
ν=1 1≤j≤rν
Recall the Abel map A at D is A(D) =
g Z X i=1
zi
ω1 , · · · ,
P0
g Z X i=1
The rank of A at D is that of the following matrix, 77
!
zi
P0
ωg
(P1 ) + O(w2 )
f1 (P1 ) .. .
(r1 −1) f1 (P1 ) f1 (P2 ) .. . (r1 −1) (P2 ) Φ = f1 .. . .. . f (P ) 1 n .. . (r1 −1) f1 (Pn )
···
fk (P1 ) .. . (r1 −1)
··· ···
fk
···
fk
(P1 ) · · · fk (P2 ) ··· .. .
(r1 −1)
··· ···
···
.. . .. .
(P2 ) · · ·
fk (Pn ) .. . (r1 −1)
fk
···
(Pn ) · · ·
fg (P1 ) .. .
(r1 −1) fg (P1 ) fg (P2 ) .. . (r1 −1) fg (P2 ) .. . .. . fg (Pn ) .. . (r1 −1) fg (Pn ))
P Denote by Φk the k-th column of this matrix, then a linear combination gk=1 ck Φk of Pgthese columns is zero if and only the order of the divisor associated with ω = k=1 is at least rν at each Pν for ν = 1, 2, · · · , n, or, equivalently, (ω) ≥ D. Hence the number of linearly independent relations between these g columns is dimH 0 (Ω−D ) = dimH 1 (X, OD ), which follows Serre duality. By Riemann-Roch theorem, dimH 1 (X, OD ) = dimH 0 (X, OD ) − 1 + g − deg(D) = dimH 0 (X, OD ) − 1 = dim|D|, hence the rank of A at D is g − dim|D|. We finally remark that the fact that the induced map j : Pic0 (X) → Jac(X) by A : Div0 (X) → Jac(X) is an isomorphism follows from our first theorem directly. Indeed, to prove the only non-trivial part of the statement, namely, the map is surjective, we note that the map A : S g (X) − Y → Jac(X) − A(Y ) is surjective, so is S g (X) → Jac(X). Therefore, for a point ξ ∈ Jac(X), its pre-image is not empty, say there exists D ∈ S g (X) mapping onto ξ. Let P0 be the base point in the definition of Abel map, then D − g · P0 is in Div0 , and which maps onto ξ. Since ξ is arbitrary, the surjectivity follows.
78
Chapter 8 Theta Functions Let Λ be the lattice embedded in Cn and we denote by T the complex torus Cn /Λ. The main concern in this chapter is to construct non-constant meromorphic function on T . This has been done for the case where n = 1 through the construction of Weierstrass P-function. This method, however, cannot be generalized to the case where n > 1. Instead, we will apply the fact that any multiply periodic function can be written as the quotient of two theta functions to reduce our task to constructing theta functions for Λ.
8.1
Reduction to Theta Functions
As a natural generalization to the definition given in last chapter, a theta function θ(z) for T = Cn /Λ is a holomorphic function in Cn satisfying the following functional equation, θ(z + λ) = θ(z) · exp{F (z, λ)} for each λ ∈ Λ, where F (z, λ) is an inhomogeneous function linear in variable z. Clearly, the divisor of θ is effective and periodic and if the divisor is zero, i.e. θ(z) has no zeros, it is a trivial theta function. The main theorem we want to prove is the following one. ˜ Theorem: Let D be an effective divisor on the complex torus T = Cn /Λ, and D n be the induced divisor on C , there exists a theta function with respect to Λ whose ˜ divisor is D. As a result of the theorem, we have following important corollary, 79
Corollary: Any multiply periodic functions can be written as a quotient of two theta functions. Proof (to the corollary): Let f be a multiply periodic function which is not identically zero. We write (f ) = D˜1 − D˜2 where D˜1 and D˜2 are effective divisors on T . From the theorem above, we can find a theta function θ whose divisor is D˜2 , then f · θ has divisor D˜1 , hence it is a holomorphic function. Since f is multiply periodic, f · θ is also a theta function. This completes our proof to the corollary. To prove the main theorem, we first note that it is simple if n = 1 as we have discussed earlier in the section on doubly periodic functions. For the general case we will need several lemmas. Lemma 1: Let U be a convex open subset of Cn and ω a d-closed 1-form on U . There exists a function on U such that df = ω. RP Proof: Fixing a point O ∈ U , we have the path integral f (P ) = O ω for any P ∈ U . To see this function R is well define, we need to show the integral is independent of the base point, i.e. γ ω = 0 for any close curve γ in U . Up to some manipulation, we may assume that ∆ is a domain inR U whose R boundary is γ. With Stokes’ theorem and the fact that dω = 0, we have γ ω = ∆ dω = 0. Lemma 2: Let U = {U1 , · · · , UN } be a covering of the complex torus T by open hypersphere and let r ≥ 0 be an integer. Let E (r) be the sheaf of differential r-forms on T , the corresponding cohomology group H 1 (U, E (r) ) is zero. Proof: Consider a collection of r-forms ωij on Ui ∩ Uj satisfying the cocycle condition, we want to show that the r-forms split, i.e. ∀ ωij on Ui ∩ Uj 6= ∅, ωij = ωi − ωj where ωi and ωj are r-forms on Ui and Uj respectively. By partition of unity to U, we get a set of non-negative smooth functions f1 , · · · , fN on T such that each fi = 0 outside Ui and at every point of T , f1 + · · · + fN = 1. Now define ωi to be f1 ωi1 + · · · + fN ωiN on Ui , we can check {ωi }1≤i≤N are the desired r-forms. Lemma 3: Let ω be a d-closed 2-form on T whose induced form on Cn is denoted by ω ˜ , x1 , · · · , x2n be the 2n real coordinates on Cn , then we have the following decomposition: ω ˜ = dψ˜ + dφ (8.1) n where ψ˜ is an induced P 1-form from a 1-form ψ on T , and φ is a 1-form on C with the expression φ = i,j ci jxi dxj . P Proof: Write ω ˜ as i,j fij dxi ∧ dxj where fij are periodic and differentiable. We have the corresponding Fourier series for fij as X (m) fij = aij · exp{2πi(m1 x1 + · · · + m2n x2n )} (8.2) m ~
80
where m ~ = (m1 , · · · , m2n ) ∈ Z2n . To satisfy the condition that ω ˜ is d-closed, we first consider every single term of ω ˜ in 2n above series and further require the series converge. For an arbitrary m ~ ∈ Z , we write X X ~ ω ˜ (m) = exp{2πi mν xν } aij dxi ∧ dxj (8.3) i,j
If m ~ = ~0, we take ψ˜ = 0 and φ = i,j aij xi dxj . Otherwise, for m ~ 6= ~0, we may assume m1 6= 0 and take it as a pivot. Rearranging terms in (8.3), we can have aij = 0 unless ~ i < j. Considering the term dx1 ∧ dxi ∧ dxj in d˜ ω (m) , we obtain the relation P
m1 aij − mi a1j + mj a1i = 0 for 1 < i < j. Substituting this into (8.3), we have, ~ ω ˜ (m) = exp{2πi
X
mν xν }
X mi a1j − mj a1i dxi ∧ dxj m 1 i,j
2n X X 1 = d{exp{2πi m ν xν } a1j dxj } 2πi · m j=2
Together we have, ω ˜=
X m ~
~ ω ˜ (m) = d{
2n X X X 1 exp{2πi mν xν } a1j dxj } + aij xi dxj 2πi · m j=2 i,j
(8.4)
as in (8.1). Now we are left to verify that the series used to define ψ˜ converge. Recall that the Fourier series in (8.2) converge to a differentiable function while the magnitude of the coefficients in (8.4) are smaller than those in (8.2), hence the series for ψ˜ represents a well-defined 1-form. As a corollary, if ω has no terms of type (0, 2) (i.e. writing (z1 , · · · , zn ) as (x1 + i · x2 , · · · , x2n−1 + i · x2n ), we do not have term dx2i ∧ dx2j for some 1 ≤ i, j ≤ in the definition of ω), then we can further require that ψ and φ be of type (1, 0). To prove the corollary, we note that in the proof of Lemma 3, φ was chosen so that dφ was equal to the constant term in the Fourier expansion of ω ˜ . With the extra condition in the corollary, φ must be chosen so that X X dφ = αij dzi ∧ dj + βij dzi ∧ d¯ zj ij
ij
81
for constants αij , βij . For this purpose, we may take X X φ= αij zi dzj − βij z¯j dzi ij
ij
which is of type (1, 0). Now let X X exp{2πi (αi zi + α ¯ i z¯i )} · (βj dzj + γj d¯ zj ) i
(8.5)
j
be a general term in the Fourier expansion of the induced form ψ˜ obtained in the proof of Lemma 3. Since ψ˜ had no constant term, we must have αν 6= 0 for some 1 ≤ ν ≤ n. Without loss of generality, we assume α1 6= 0. Since the derivative of (8.5) has no terms of type (0, 2), we have the identity α ¯ 1 γj = α ¯ j γ1 . Thus, if we abstract ! X γ1 d exp{2πi (αi zi + α ¯ i z¯i )} 2πi¯ α1 i from (8.5), we will obtain a form of type (1, 0) without changing the derivative. The subtraction will at most double the coefficients in (8.5), so the modified ψ˜ still has the differentiability required. With these lemmas, we can proceed to prove the main theorem. Proof of the theorem: First we note that for an arbitrary covering {Uα } of the complex torus T = Cn /Λ, it can be refined to a finite covering by open hyperspheres. Indeed, to each point P of T , we can choose a Uα containing P and an open hypersphere containing P and is contained in this Uα . The set of all these open hyperspheres forms a covering of the torus as a refinement to the original covering. Since T is compact, we can select from this a finite sub-covering. Now we assume that the effective divisor D has a finite description {Ui , φi } in which each Ui is an open hypersphere in T . Thus φi /φj is holomorphic and nowhere zero on Ui ∩ Uj and so the form ωij = d (log(φi /φj )) on Ui ∩ Uj are of type (1, 0) satisfying the cocycle relation. Hence by Lemma 2, ωi j split, i.e., there are 1-forms ωi of type (1, 0) on Ui such that ωi − ωj = ωij = d (log(φi /φj )) (8.6) on Ui ∩ Uj . And dωi = dωj on Ui ∩ Uj , so the dωi piece together to give a locally closed 2-form on T with no terms of type (0, 2). Then we can use Lemma 3 and its corollary to write ω ˜ as dψ˜ + dφ. Now we write ψi = ωi − ψ on Ui ; thus ψi are forms of type (1, 0) such that dlog(φi /φj ) = ψi − ψj on Ui ∩ Uj . The pre-image of Ui in Cn is the disjoint union of open hyperspheres Uiλ where λ runs through elements of Λ and Uiλ is the translation of Ui0 by λ. Moreover d(ψ˜i − φ) = 0 82
on Uiλ and hence by Lemma 1, there is a function fiλ on Uiλ such that diλ = ψ˜i − φ on Uiλ . Since the right hand side of the identity is of type (1, 0), fiλ is holomorphic on Uiλ and dfiλ − dfjµ = ψ˜i − ψ˜j = dlog(φ˜i /φ˜j ) on Uiλ ∩ Ujµ . Hence φ˜i e−fiλ is a constant multiple of φ˜j e−fjµ in the intersection, both being holomorphic. By analytic continuation of φ˜1 e−f1λ , we obtain a holomorphic ˜ function θ(z) in the whole Cn whose divisor is D. It remains to prove θ(z) is a theta function for Λ. For this purpose, we fix a λ ∈ Λ and suppose z ∈ U10 ; then z + λ ∈ U1λ . By the periodicity of φ˜1 we see θ(z + λ) = C · exp{f10 (z) − f1λ (z + λ)} θ(z) P for some non-zero constant C. Writing φ = ij (aij zi + bij z¯i )dzj , we obtain X θ(z + θ) dlog = df10 (z) − df1λ (z + θ) = φ(z + λ) − φ(z) = (aij zi + bij z¯i )dzj θ(z) ij follows where λi are the complex coordinates of λ. The functional equation for θ(z+θ) θ(z) by integration, which completes our proof. As a result, an effective divisor determines a theta function up to multiplication by some trivial theta function.
8.2
Theta Functions and Riemann Forms
In this section we will focus on the functional equation for theta functions, namely, θ(z + λ) = θ(z) · exp{F (z, λ)} ∀ λ ∈ Λ
(8.7)
In this and next section, we will study the necessary and sufficient condition for (8.7) to have non-trivial solutions, which turns out to be the existence of a positive semidefinite non-trivial Riemann form. Its necessity will be proved in this section while the sufficiency will be shown in the next section. Let λ, λ0 be two elements of Λ, from the identity θ(z + λ + λ0 ) θ(z + λ + λ0 ) θ(z + λ) = · θ(z) θ(z + λ) θ(z) we obtain F (z, λ + λ0 ) ≡ F (z + λ, λ0 ) + F (z, λ) mod 2πi
(8.8)
Recall that F (z, λ) is inhomogeneous linear in the variable z, now we can split it into two parts, say F (z, Λ) = 2πi{L(z, λ) + J(λ)} 83
such that L(z, λ) is linear and homogeneous in z while J(λ) is constant for z. Substituting this into (8.8), we have L(z, λ + λ0 ) = L(z, λ) + L(z, λ0 ) + a
(8.9)
J(λ + λ0 ) − J(λ) − J(λ0 ) = L(λ, λ0 ) + b
(8.10)
where a, b are constant integers. However, we note in (8.9), setting z to 0, by linearity, we must have a = 0, hence L(z, λ + λ0 ) = L(z, λ) + L(z, λ0 ). Since Cn is viewed as real vector space spanned by Λ, i.e. Cn = Λ ⊗ R, this implies L can be extended to a function on Cn × Cn , C-linear in its first variable and R-linear in the second one. We define two Riemann forms as following, E(z, w) = L(z, w) − L(w, z) (8.11) H(z, w) = E(iz, w) + iE(z, w)
(8.12)
They have the following fundamental properties, Theorem: E is a R-bilinear, real-valued and alternating form on Cn × Cn and E(z, w) = E(iz, iw). It follows that the induced form H is Hermitian. Moreover E is integer-valued on Λ × Λ. Proof: Since L is R-linear in both variables, it is clear that E is R-bilinear. It is alternating which follows (8.11). In (8.10), the left hand side is symmetric in λ and λ0 , thus L(λ, λ0 ) ≡ L(λ0 , λ)mod 1. Therefore E is integer-valued on Λ × Λ, and by the real-bilinearity, E must also be real-valued on Cn × Cn . Moreover, by the C-linearity in its first variable, we have, E(z, w)−E(iz, iw) = L(z, w)+L(iw, iz)−L(w, z)−L(iz, iw) = i{E(w, iz)−E(z, iw)} In the identity above, the left hand side is real-valued while the right hand side is pure imaginary-values, hence both of them shall vanish, i.e E(z, w) = E(iz, iw) and E(z, iw) = E(w, iz). For H, clearly it is linear in the first variable; for some α ∈ R, H(z, αw) = E(iz, αw) + iE(z, αw) = α{E(iz, w) + iE(z, w)} = αH(z, αw) and for β = iα H(z, βw) = E(iz, βw) + iE(z, βw) = E(iz, iαw) + iE(z, iαw) = αE(z, w) − iαE(iz, w) = −βE(iz, w) − βiE(z, w) = −βH(z, w) By linearity, we see that H is conjugate linear in its second variable. E is called an alternating Riemann form on Cn × Cn with respective to Λ. The corresponding H is called a Hermitian Riemann form. We will show that H associated 84
with a theta function must be positive semi-definite. In particular, if H = 0, the corresponding theta function is trivial. Before proceeding on, we will have some remark for the case n = 1 where this condition is automatically satisfied so that everything is reduced to the study of doubly periodic functions which we have discussed earlier. In the case that n = 1, let λ1 , λ2 be a basis for Λ, then we have, 1 2πi
Z ∂Γ
1 θ0 (z) dz = { θ(z) 2πi
Z
z0 +λ1
Z
z0 +λ2
+ z0 +λ1 +λ2
z0
θ0 (z) dz} θ(z)
Z z0 +λ1 +λ2 Z z0 θ0 (z) 1 + + { dz} 2πi z0 +λ2 z0 +λ1 θ(z) = L(λ1 , λ2 ) − L(λ2 , λ1 ) = E(λ1 , λ2 ) = m where m is the number of zeros of θ in Γ, which is a positive integer. Since E is Rbilinear and alternating, the value of E(z, w) on C × C is completely determined by the value of E(λ1 , λ2 ). Meanwhile, the ratio E(z, w)/A(z, w) = E(iz, iw)/A(iz, iw) is a real constant, where A(z, w) is the area of the parallelogram generated by Oz and Ow and A(iz, iw) is the area of the same parallelogram after a rotation by π2 , hence we have E(z, w) = E(iz, iw). Consequently, given a lattice Λ, and the number of zeros of θ in Γ, there is a unique alternating Riemann form, which is non-zero as long as the corresponding theta function is non-trivial. However, in the case that n > 1, it is in general not true that non-zero Riemann forms exist. Let λ1 , · · · , λ2n be a basis for Λ, then E, as a R-bilinear alternating form is determined by E(λi , λj ), which are integers. The additional condition that E(z, w) = E(iz, iw) induces linear relations with real coefficients between E(λi , λj ), which in general have no non-trivial solutions (We will discuss this in detail later). It is remarkable, however, that with the obvious topology, the set of lattices Λ (for a fixed n > 1) which admits non-zero Riemann forms is dense in the set of all lattices. Now, substituting K(λ) = J(λ) − 21 L(λ, λ) into (8.10), we get 1 K(λ + λ0 ) − K(λ) − K(λ0 ) ≡ E(λ, λ0 ) mod 1 2
(8.13)
Viewing E as a skew-symmetric matrix with respect to a basis for Λ, it is possible to write E(z, w) = B(z, w) − B(w, z), where B is R-bilinear on Cn × Cn and is integer-valued on Λ × Λ. Writing K 0 (λ) = K(λ) − 21 B(λ, λ), then we have K 0 (λ + λ0 ) ≡ K 0 (λ) + K 0 (λ0 ) mod 1
(8.14)
from (8.13). This implies K 0 is a group homomorphism from Λ to C/Z. Two theta functions are said to be equivalent if their quotient is a trivial theta function (recall the definition, a trivial theta function has no zeros in Λ). From last section, we 85
know that for every effective divisor D on a complex torus T = Cn /Λ, there is a theta function with respect to Λ whose divisor is D. Together we see that each effective divisor on Cn /Λ corresponds to an equivalence class of theta functions. Moreover, we will see that given an E, all the L that is compatible with (8.11) can be obtained with an equivalence class of theta functions. To see this, we consider an arbitrary theta function θ1 . Now we obtain another theta function θ2 by θ2 (z) = θ1 (z) · exp{2πi[Q(z, z) + R(z) + S]}
(8.15)
where Q is C-bilinear symmetric, R is C-linear and S is a constant. θ2 is equivalent to θ1 since f (z) = exp{2πi[Q(z, z) + R(z) + S] is a trivial theta function, whose functional equation is f (z + λ) = exp{2πi[Q(z + λ, z + λ) + R(z + λ) + S]} = f (z) · exp{2πi[2Q(z, λ) + Q(λ, λ) + R(λ)]} Hence ∆F (z, λ) := Fθ2 − Fθ1 = 2πi[Q(z, λ) + Q(λ, λ) + R(λ)], i.e ∆L(z, λ) = Lθ2 − Lθ1 = 2Q(z, λ) and ∆J(λ) = Jθ2 − Jθ1 = Q(λ, λ) + R(λ). ∆K = Kθ2 − Kθ1 = R(λ), so is the change of K 0 . In particular, since Q is symmetric, in this case, we have ∆E(z, w) = Eθ2 − Eθ1 = L(z, w) − L(w, z) = 2Q(z, w) − 2Q(w, z) = 0. Hence for theta functions in the same class, the quantity E associated is unchanged; their quotients induce a set of L which vary by addition of C-bilinear symmetric functions. Moreover, for a given E, suppose a certain L is compatible with (8.11), then any L0 in the form L0 = L + Q with Q being C-bilinear symmetric will still be compatible with (8.11); conversely, the difference between two L’s satisfying (8.11) must be a Cbilinear symmetric function. In summary, (1) there is an one-to-one correspondence between a Riemann form E and a class of theta functions; (2) all the L that are compatible with (8.11) with respect to this E are determined by a class of equivalent theta functions (the determination is not class-specific though). In each equivalence class we can choose a normalized one, which is unique up to multiplication by a constant. In the following part, we will give a definition of normalized theta functions; their existence follows directly as our definition will give an explicit example. We will then prove that with respect to a certain equivalence class, the normalized theta function is unique up to multiplication. From (8.12), we have H(z, w) = E(iz, w) + iE(z, w) = −E(z, iw) + iE(z, w) and H(w, z) = E(iw, z) + iE(w, z) = −E(w, iz) − iE(z, w). Subtracting the latter from the first one, we get 1 1 iH(w, z) − iH(z, w) = E(z, w) 2 2 Hence for a fixed Riemann form E that specifies an equivalence class, we can choose L(z, w) = − 12 iH(z, w). Let f (λ) be the imaginary part of K(λ), (8.13) indicates f (λ + λ0 ) = f (λ) + f (λ0 ) on Λ. Hence it can be extended to a R-linear real-valued function on Cn . In (8.15), we define R(z) = −f (iz) − if (z), whose C-linearity follows 86
that R(iz) = −f (i2 z) − if (iz) = f (z) − if (iz) = iR(z). Therefore K 0 = K(λ) + R(λ) is real-valued. We say that a theta function θ is normalized if L(z, w) = − 21 iH(z, w) and K and K 0 are both real-valued. To prove the uniqueness, suppose θ1 and θ2 are both normalized compatible with (8.15), then ∆L = 2Q(z, λ) = 0 and ∆K = ∆K 0 = R(λ) is real-valued. i.e f = Im(R) = 0, thus R(z) = −f (iz) − if (z) = 0. Together, we have θ2 = θ1 exp{2πi · S}. Since each effective divisor on Cn /Λ corresponds to an equivalence class of theta functions, we can associate to each of them an alternating Riemann form E. This map extends to a homomorphism from the group of divisors on Cn /Λ to the additive group of Riemann forms with respective to Λ. In particular, we will see that the kernel of the homomorphism is the zero divisors, which corresponds to the class of trivial theta functions. One direction of this statement is obvious, i.e. for a trivial theta function θ(z), E is zero. Recall, any trivial function has the form exp{Q(z)} for some quadratic polynomial Q(z), hence, we may write θ = az 2 + bz + c where a, b, c ∈ C. From the functional equation for theta functions we obtain F (z, λ) = 2aλz +aλ2 +bλ, hence L(z, λ) = 2aλz and E(z, λ) = L(z, λ) − L(λ, z) = 0. The other direction will be proved after the next theorem, which we have mentioned at the beginning of this section. Theorem: The Hermitian Riemann form H associated with a theta function θ(z) is positive semi-definite. Proof: To see this, we pick up a normalized theta function θ(z). Since H is determined by a class of theta functions, the value will be invariant if we choose another one in the same class as θ(z). Writing φ(z) = θ(Z)exp{− π2 H(z, z)}, we have φ(z + λ) = φ(z)exp{πi[E(z, λ) + 2K(λ)]}
(8.16)
from the functional equation θ(z + λ) = θ(z)exp{πH(z, λ) +
π H(λ, λ) + 2πi · K(λ)} 2
Since E and K are real-valued, φ(z) is periodic, hence |φ(z)| must be bounded, say by some constant C. Therefore |θ(z)| ≤ C ·exp{ π2 H(z, z)}. Suppose there is some z0 such that H(z0 , z0 ) is negative, we write z = t · z0 with t ∈ C. Now θ(t · z0 ) is holomorphic 2 viewed as a function in t, and it follows |θ(t · z0 )| ≤ C · exp{ π|t| H(z0 , z0 )}, which goes 2 to zero as |t| goes to ∞. Hence, by Liouville’s theorem, θ(t · z0 ) is a constant function in t which vanishes identically in C. In particular, θ(z0 ) = 0. There is a neighborhood U of z0 in which H(z, z) < 0, so in the same way we have θ(z) = 0 in U . By analytic continuation, θ(z) is identically zero, which is a contradiction. This concludes our proof. A theta function is called degenerate if the associated H is not positive definite. We will show that a degenerate theta function is essentially the same as the one 87
on a quotient space Cn /W , where W is defined to be the set of w ∈ Cn such that H(z, w) = 0 for any z ∈ Cn , called the kernel of H. Clearly W forms a vector space over C. Lemma: Let H be a positive semi-definite Hermitian Riemann form on the vector space Cn , and W be its kernel. Then W consists of all w such H(w, w) = 0 and the H induces a positive definite Hermitian form on quotient space Cn /W . Moreover, the image of Λ under this projection is also a lattice in Cn /W . Proof: It is trivial that any w ∈ W satisfies H(w, w) = 0. For the other direction, suppose H(w, w) = 0, then for any t ∈ C and any v ∈ Cn , we have 0 ≤ H(tw + v, tw + v) = 2 · Re{tH(w, v)} + H(v, v) For a fixed v, this holds for all t if and only if H(w, v) = 0. Since the v is arbitrary, we see that w ∈ W . Now for any w1 , w2 ∈ W and any v1 , v2 ∈ Cn , we have H(w1 +v1 , w2 +v2 ) = H(v1 , v2 ), so H induces a positive definite Hermitian form in Cn /W . Since Λ spans Cn , it is clear that it also spans Cn /W and to ensure that the image of Λ is a lattice, it remains to verify that the image is discrete in the quotient space. Let λ1 , · · · , λ2n be a basis of Λ and N be a neighborhood of the origin of Cn /W such that ∀ z ∈ N , |H(z, λi )| < 1 for i = 1, 2, · · · , 2n. If λ ∈ Λ in N , then |E(λ, λi )| = |ReE(λ, λi )| < 1. Since E is integer-valued on Λ × Λ, E(λ, λi ) = 0, thus E(λ, z) = 0 for any z ∈ Cn by the R-linearity of E in its second variable. Therefore H(λ, z) = 0 for any z ∈ Cn , i.e. λ ∈ W . This shows we can find a neighborhood of origin in Cn /W which contains only one point of the image of Λ, namely, the origin itself. Hence the image of Λ is discrete in Cn /W , which becomes a lattice. Now we are ready to connect theta functions and Hermitian Riemann forms in the quotient space Cn /W . Theorem: Let θ(z) be a normalized theta function on Cn with respect to the lattice Λ. Let H be the associated Hermitian Riemann form whose kernel is W . Denoted by Λ0 the image of Λ in the quotient space Cn /W , then θ induces a function θ0 in Cn /W , which is a theta function with respect to Λ0 . Moreover the induced Hermitian Riemann form on Cn /W corresponding to θ0 is just the one induced by H. Proof: It suffices to check that θ(z) induces a well-defined function in the quotient space, namely, it is constant on each coset of W and the rest of the theorem follows directly. For z ∈ Cn and w ∈ W , we have, from the proof to the last theorem, |θ(z + w)| ≤ C · exp{ π2 H(z + w, z + w)} for some θ-dependent constant. But w is in W , hence H(z + w, z + w) = H(z, z), and |θ(z + w)| ≤ C · exp{ π2 H(z, z)}. Viewed as a function in w, θ(z + w) is holomorphic and bounded, hence by Liouville’s theorem, it is constant. In particular, θ(z + w) = θ(z), which completes our proof. As a corollary of this theorem, we see that if H is identically zero, or equivalently, 88
E is identically zero, then W = Cn , which implies the normalized theta function associated to E is constant, i.e. the corresponding equivalence class is the set of all trivial theta functions. Now suppose that the Hermitian Riemann form H is non-degenerate, or, equivalently, E is non-singular. As an integer-valued alternating form on Λ × Λ, E is represented by a non-singular skew-symmetric matrix with elements in Z for a certain basis of Λ; and the determinant of this matrix is a perfect square independent of the choice of the basis. The positive square root of the determinant is called the Pfaffian of E, denoted by Pf(E). By suitable choice of basis, the matrix representing E can transformed into the following canonical form 0 D E= −D 0 where D = diag(d1 , · · · , dn ), and di are positive integer such that di divides di+1 for i = 1, 2, · · · , n − 1. Then Pf(E) = d1 d2 · · · dn . The di are uniquely determined by E and are independent of the choice of basis for Λ. A theta function for Λ will also be a theta function for any sub-lattice Λ1 ⊂ Λ and its Pfaffian with respect to Λ1 will be [Λ : Λ1 ] times the Pfaffian with respect to Λ. On the other hand, it may also be a theta function with respect to Λ2 ⊃ Λ, but if it is non-degenerate, there can only be finitely many such Λ2 for we must have that [Λ2 : Λ] divides the Pfaffian with respect to Λ.
8.3
Construction of Theta Functions
In this section we will construct theta functions corresponding to given L and J satisfying (8.9), (8.10), and Riemann forms E, H compatible with (8.11), (8.12). Moreover, it suffices to consider the case when H is positive definite for otherwise we may take a quotient over the kernel W as we did in last section. Theorem(Frobenius): Suppose L(z, w) is C-linear in its first variable and R-linear in the second variable. J satisfies (8.10) while E, H are defined in (8.11) and (8.12) respectively, with E being R-bilinear, real-valued alternating on Cn × Cn and integervalued on Λ×Λ, and H being Hermitian, positive definite. Then the space of functions θ(z) holomorphic on Cn satisyfing θ(z + λ) = θ(z)exp{2πi[L(z, λ) + J(λ)]}
(8.17)
for each λ ∈ Λ has dimension equal to the Pfaffian of E, denoted by Pf(E). Such holomorphic theta functions are called to be of type (L, J). Proof: We choose a basis λ1 , · · · , λ2n for Λ with respect to which E is represented by 89
the matrix
E=
0 D −D 0
where D = diag(d1 , · · · , dn ) and the di are positive integers. We will show that λ1 , · · · , λn are linearly independent over C. Suppose, on the contrary, they are linearly dependent over C, there is a relation (a1 + ib1 )λ1 + · · · + (an + ibn )λn = 0 P P with ai , bi real and not all zero. Thus we have ni=1 ai λi = α and ni=1 bi λi = −iα for some α ∈ Cn . Since λ1 , · · · , λn are linearly independent over R, α 6= 0. But on the other hand E(λi , λj ) = Eij = 0 for 1 ≤ i, j ≤ n, therefore we have X 0 < H(α, α) = E(iα, α) = ai bj E(λi , λj ) = 0 ij
which is impossible. So λ1 , · · · , λn are linearly independent over C and we take them to be a basis for the coordinate system on Cn . Note that multiplying θ(z) by exp{2πi[Q(z, z)+R(z)]} for some C-bilinear symmetric Q and some C-linear R, we have L(z, P w) increased by 2Q(z, w) and P K(λ) increased 1 z w L(λ , λ ), R(z) = by R(λ). We choose Q(z, w) = − i jP i zi K(λi ) where ij i j 2 P P wP = j wj L(z, λj ) and ∆K(z) = i zi λi , then ∆L(z, w) = − i wi λi and z = − i zi K(λi ), which implies L(z, λi ) = K(λi ) = 0 and thus J(λi ) = 0, for i = 1, 2, · · · , n. The modified θ(z) turns out to be periodic in each of zi with period one, thus we can expand it as a multiple Fourier series, X θ(z) = c(m1 , · · · , mn )exp{2πi(m1 z1 + · · · + mn zn } Since (8.9) and (8.10) are satisfied, the functional equation (8.17) holds for all λ ∈ Λ if it holds for λ1 , · · · , λ2n . For the first n of them, we have proved already. For indices µ, ν between 1 and n, we have E(λν+n , λµ ) = L(λν+n , λµ ) − L(λµ , λν+n ) where L(λν+n , λµ ) = 0 and L(λµ , λν+n ) = dν if µ = ν and equals zero otherwise. Therefore, by C-linearity of L in its first variable, we have L(z, λν+n ) = P (ν) (ν) µ zµ L(λµ , λν+n ) = dν zν . In particular, L(λµ+n , λν+n ) = dν λµ+n , where λµ+n is the ν-th coordinate of λµ+n . Now the functional equation (8.17) takes the form θ(z + λν+n ) = θ(z)exp{2πi[dν zν + J(λν+n )]} which is equivalent to X (1) (n) c(m1 , · · · , mn )exp{2πi[m1 (z1 + λν+n ) + · · · + mn (zn + λν+n )]} =
X
c(m1 , · · · , mν − dν , · · · , mn )exp{2πi[(m1 z1 + · · · + mn zn ) + J(λν+n )]} 90
or equivalently, n X c(m1 , · · · , mν − dν , · · · , mn ) (µ) = exp{2πi[ mµ λν+n − J(λν+n )]} c(m1 , · · · , mn ) µ=1 n X L(λν+n , λµ+n ) = exp{2πi[ mµ − J(λν+n )]} dµ µ=1
Formally, this means that we can choose the c(m1 , · · · , mn ) with 0 ≤ mν < dν for each index ν and all the other coefficients are determined by the recursive relation given above. This gives d1 d2 · · · dn degrees of freedom, which is equal to the Pfaffian of E. It remains to check that the resulting formal Fourier series converge. We note, provided each mν stays in a fixed congruence class mod dν , the coefficients are c(m1 , · · · , mn ) = exp{−πiL(β, β) + γ(mν ) + C} P mν where β = λ , γ is a linear form in mν and C is a constant. We note that dν ν+n exp{γ(mν ) + C} is just a multiplication by constant to the terms in the Fourier series, which has no effect on whether the series converge, thus it suffices to show the imaginary part of L(z, z) is negative definite. We write z = x + iy where x and y are linear combinations of λ1 , · · · , λn over R. Then, L(z, z) = L(x, z)+iL(y, z) = E(x, z)+L(z, x)+iE(y, z)+iL(z, y) = E(x, z)+iE(y, z) By the R-linearity of L in its second variable and the fact that L(z, λi ) for i = 1, 2, · · · , n. Since E is real-valued, the imaginary part of this is Im(L(z, z)) = E(y, z) = E(y, x)+E(y, iy) = −E(iy, y) = −E(iy, y)−iE(y, y) = −H(y, y) < 0 for y 6= 0. Since z lies in the real vector space spanned by λn+1 , · · · , λ2n , we cannot have y = 0 for non-zero z, since it would otherwise imply a linear dependence over R among λ1 , · · · , λ2n , which is impossible. This completes the proof of the theorem. Under the condition of the theorem, almost all of the theta functions for Λ will not be theta functions for any lattice Λ0 strictly containing Λ, for if we let m = [Λ0 : Λ] and d be the Pfaffian of E with respect to Λ, then if there is theta function satisfying the same functional equation (8.17), m must divide d. So there are only finitely many candidates for Λ0 . To each of them corresponds only finitely many ways of extending the functional equation (8.17) to Λ0 , and to each of these extensions corresponds a space of solutions of dimension d/m. So the theta functions for a fixed Λ0 form a finite set of subspaces, each of dimension d/m. Two linearly equivalent divisors D1 , D2 differ by a principal divisor, say (f ) for some meromorphic function f on T . Recall that (f ) can be decomposed into two effective divisors, say D10 − D20 . Suppose the holomorphic theta function θ2 (z) corresponds to D20 is of type (L, J), then the theta function θ1 = f · θ2 corresponding to D20 satisfies 91
the same functional equation, hence is also of type (L, J). This implies the Riemann forms associated with θ1 and θ2 are the same, i.e. any meromorphic function f can be written as the quotient of two holomorphic theta functions of the same type. Recall we have defined a monomorphism from the group of effective divisors on T into the group of Riemann forms. Now we extend the homomorphism such that it maps the group of divisors on T into the group of Riemann forms in the natural way, then principal divisors are mapped onto 0, which implies the Riemann forms associated with two linearly equivalent divisors are the same. let D be an effective divisor on T , we define a space of meromorphic function as following L[D] = {f ∈ M(T )| (f ) + D ≥ 0} (8.18) Let θ0 be a holomorphic theta function whose divisor is D, we define L(θ0 ) to be the space of all holomorphic theta functions of the same type as θ0 , then for any θ ∈ L(θ0 ), θ 7→ θ/θ0 gives an isomorphism from L(θ0 ) onto L[D]. By the Frobenius theorem, we have dimL[D] = dimL(θ0 ) = Pf(E), where E is the Riemann form associated with L[D]. Lemma: Suppose D0 , D1 , · · · , Dm are effective divisors on T = Cn /Λ and at least D0 is non-degenerate. Then there is a polynomial P of degree n such that m X dimL( rj Dj ) = P (r0 , r1 , · · · , rm )
(8.19)
j=0
whenever rj ≥ 0 for all j and r0 > 0. Proof: For each Dj , let θj be the corresponding holomorphic theta function,Qand Hj P rj r H corresponds to m be the associated Hermitian Riemann form. Then m j=0 θj j=0 j j and is positive definite. This implies m m X X r0 r1 rm dimL( rj Dj ) = dimL(θ0 θ1 · · · θm ) = Pf( rj Ej ) j=0
j=0
which is a homogeneous polynomial of degree n in each rj . An effective divisor is called degenerate (res. non-degenerate) if the associated Riemann form is deg In this section, we are going to prove the following main theorem. Theorem: Suppose that there is a non-degenerate effective divisor on complex torus T = Cn /Λ, then the field of meromorphic functions on T is finitely generated of transcendence degree n over C. To prove the theorem, we first prove that the transcendence degree of the field of meromorphic functions on T denoted by M(T ) over C is at most n, denoted by trdegC M. And if the upper bound is attained, M(T ) is finitely generated over C. 92
Proof: Let f1 , · · · , fn+1 be meromorphic functions on T and let D be an effective nondegenerate divisor on T such that (fi ) + D ≥ 0 for each i. The number of distinct monomials in fi of total degree at most N is N +n+1 (N + n + 1)! N n+1 = = + O(N n ) (8.20) n+1 N !(n + 1)! (n + 1)! All these monomials lie in the space L(N D), whose dimension is a polynomial in of degree n in N , namely dimL(N D) ∈ O(N n ). Hence when N is sufficiently large, (8.20) >> dimL(N D), and there is some linear dependence relation among these monomials, which is an algebraic relation among f1 , · · · , fn+1 . This shows trdegC M is at most n. To prove that the field is finitely generated when trdegC M = n, we take f1 , · · · , fn to be n meromorphic functions on T , algebraically independent over C. Let D0 be an effective non-degenerate divisor on T such that (fj )+D0 ≥ 0 for each j. Let Pf(D0 ) = Qn i=1 di be the Pfaffian of the Riemann form associated with D0 . Write M = n!Pf(D0 ). Let g be an arbitrary meromorphic function on T such Qn that (g) + D1 ≥ 0 for some effective divisor D1 on T whose Pfaffian is Pf(D1 ) = i=1 bi . Then we have n n Y Y n dimL(N D0 +M D1 ) = (N ·di +M ·bi ) = N di +O(N n−1 ) = N n ·Pf(D0 )+O(N n−1 ) i=1
i=1
(8.21) as N → ∞. On the other hand, the space L(N D0 + M D1 ) contains all the monomials in the form P m1 mn m f1 · · · fn g with m1 , · · · , mn non-negative, mi ≤ N and 0 ≤ m ≤ M . The number of such monomials is 1 N +n (N + n)!(M + 1) = (Pf(D0 ) + )N n + O(N (n−1) ) (8.22) (M + 1) = N !n! n! n so when N is large enough, (8.22) (8.21), which implies that there is some linear dependence relation among these monomials. Since f1 , · · · , fn are assumed to be algebraically independent, this relation must have g involved. This means g is algebraic of degree at most M = n!Pf(D0 ) over C{f1 , · · · , fn }. Since C{f1 , · · · , fn } is of characteristic zero, by primitive element theorem, we can pick our g to a primitive element so that M(T ) = C{f1 , · · · , fn }(g). Now it is clear that M(T ) is algebraic of degree at most M over C{f1 , · · · , fn }. We complete proving the theorem by showing that there exist a set of meromorphic functions f1 , · · · , fm such that the extension C(f1 , · · · , fm ) has transcendence degree at least n over C. It then follows trdegC (M(T )) is exactly n. Let D be a positive non-degenerate divisor on T whose Hermitian Riemann form denoted by H(z, w) is positive definite. We take f1 , · · · , fm to be a basis for L(3D), 93
∂fi then it suffices to show the matrix ( ∂z )ij for 1 ≤ i ≤ m and 1 ≤ j ≤ n has rank n at j any point on T outside the support of D (so that the denominators will not vanish). Assume that, on the contrary, there is a point w ∈ / Supp(D) at which the matrix has rank less than n. We can apply a linear transformation on zi so that
∂f = 0 at z = w ∂z1
(8.23)
for each f ∈ L(3D). Let θ(z) be a theta function corresponding to D, then for any u, v ∈ Cn , we obtain a meromorphic function on T : f (z) =
θ(z − u)θ(z − v)θ(z + u + v) θ(z)3
(8.24)
Let D1 be an effective divisor corresponding to the theta function θ(z −u)θ(z −v)θ(z + 1 ∂θ u + v), then we have (f ) + 3D = D1 ≥ 0, therefore f ∈ L(3D). Writing θ(z) =φ ∂z1 and applying our assumption (8.23) to f , we have φ(w − u) + φ(w − v) + φ(w + u + v) − 3φ(w) = 0
(8.25)
If we denote φ(w − u) − φ(w) by g(u) as a function in u, (8.25) becomes g(u) + g(v) + g(−u − v) = 0 with g(0) = 0. Choosing v = 0, g(u) = −g(−u), i.e. g is odd, whence we have g(u) + g(v) = −g(−u − v) = g(u + v), i.e. the function g(u) is additive, linear over Q. Since g is meromorphic, the linearity P even extends to be over C. Multiplying θ(z) by P a trivial theta function exp{2πi ij aij zi zj }, we can have g(u) increased by −4πi j a1j zj . So it is possible to replace the original θ(z) by an equivalent theta function so that g(u) = 0. Then we have ∂θ(u) ∂ (θ(u) · e−u1 φ(w) ) = [ − φ(w)θ(u)] · e−u1 φ(w) = g(u)θ(u) · e−u1 φ(w) = 0 ∂u1 ∂u1 which means θ(u) · e−u1 φ(w) does not depend on u1 . This implies that the Hermitian Riemann form H(z, z) associated with θ(u)exp{−u1 φ(w)}, which is the same one for θ, vanishes if 0 6= z ∈ Cn takes the form (1, 0, · · · , 0). This contradicts the condition that H is positive definite. This completes our proof to the main theorem. In the general case, let W be the intersection of the kernels of all the positive semidefinite Hermitian Riemann forms on Cn , then there is a positive semi-definite Hermitian Riemann form on Cn whose kernel is precisely W , so that Cn /W and the associated lattice define a torus T 0 on which there is a non-degenerate effective divisor so that our main theorem applies. It follows from the main theorem in last section that every meromorphic function on a torus T is induced by a meromorphic function on T 0 , so the theory for T is essentially the same as that for T 0 . The following theorem states that under certain condition, a complex torus can be embedded into the projective space. 94
Theorem: Let D be a non-degenerate effective divisor on T = Cn /Λ, the L(3D) induces an embedding of T into projective space as a complete non-singular variety. Proof: We take a positive non-degenerate divisor on T , by the main theorem, we can find a basis of n meromorphic functions f1 , · · · , fn for the space L(3D) which also generate the field of meromorphic functions on T . Let θ(z) be a theta function corresponding to D, and f be the function defined in (8.24). The embedding ι from T into PN is defined by, ι : z 7→ (f1 (z)θ3 (z), · · · , fm (z)θ3 (z)) for any z ∈ Cn , each of these coordinates being holomorphic in Cn . Since for every u,v in Cn , f (z) · θ(z)3 = θ(z − u) · θ(z − v) · θ(z + u + v) (8.26) is a linear combination of these coordinates, there is no z ∈ Cn at which they all vanish, i.e., there are no base points so our map is well defined. To see that ι separates points, we consider z1 ,z2 ∈ Cn such that z1 6≡ z2 mod Λ, we can choose u and v so that the function (8.26) vanishes at z1 but not at z2 . For example, we choose u so that z1 − u is in the support of D while z2 − u is not, and we choose v in general positions. ∂fi )ij has full rank at any points Recall the proof of the main theorem, the matrix ( ∂z j not on the support of D, image of ι is non-singular outside the support of D. A similar argument works for the points on the support of D. This implies that the image lies in a complete variety of dimension n. Since the image is compact and we have seen that it is a complex manifold of dimension n, it must be the whole variety. This completes our proof.
Conversely, if a effective divisor on T is degenerate, there are infinitesimal translations leaving it invariant, and every effective divisor linearly equivalent to it has the same Riemann form and therefore is invariant under the infinitesimal translations. So the map into projective space induced by such a divisor would not even preserve dimension which is absurd. Hence if T can be regarded as an algebraic variety, it must contain a non-degenerate effective divisor. Moreover, the projective embedding of T carries with it the group structure on T , so the associated projective variety is an abelian variety and for this reason, T is called an abelian manifold, denoted by A if T contains a non-degenerate effective divisor. In summary, we have, for a complex torus T , the following three equivalent conditions: (1)T is a manifold of complex points on abelian variety; (2)T contains a non-degenerate effective divisor; (3)It possesses a non-degenerate Hermitian Riemann form. As a remark, we note that the first part of our proof of the main theorem does not depend on the existence of a non-degenerate divisor on the complex torus. Actually, 95
there is a more general theorem of by Siegel about the size of the field of meromorphic functions on a compact complex manifold. Theorem(Siegel): The meromorphic function field M(X) on a compact complex manifold X has transcendence degree ≤ dim(X). To prove the theorem, we first introduce two lemmas. Lemma 1: If f and g are holomorphic functions at a point x ∈ Cn and are relatively prime as elements of Ox = C{z1 , · · · , zn }, there there exists a neighborhood U of x such that f and g are holomorphic in U and are relatively prime as elements of Oy for every y ∈ U . Proof of the lemma: Multiplying f and g by invertible elements of Ox and using Weierstrass preparation theorem, we can arrange that f and g are monomials in z1 over C{z2 , · · · , zn }. Since they are relatively prime, there exist u,v ∈ C{z1 , · · · , zn } such that f · u + g · v = r with r ∈ C{z2 , · · · , zn } in a neighborhood U of x. Suppose that f and g have a common factor h ∈ Oy for some y ∈ U , then h|r and applying Weierstrass preparation theorem again, we see that h is an invertible element of Oy times an element h ∈ C{z2 , · · · , zn }. But h1 |f and since f is a monomial in z1 over C{z2 , · · · , zn }, it follows that h1 is invertible in C{z2 , · · · , zn }. Lemma 2(Schwarz): Let f (z) = f (z1 , · · · , zn ) be a holomorphic function in the polydisk defined by |zi | ≤ 1 for i = 1, · · · , n, and suppose that M = max|zi |≤1 |f (z)|. Writing m0 for the maximal ideal of the local of analytic functions at the origin. If f ∈ mh0 , that is, if f and all its derivatives of degree ≤ h − 1 vanish at origin, then |f (z)| ≤ M · max |zi |h i
(8.27)
for z in the open polydisk |zi | < 1, i = 1, 2, · · · , n. Proof of the lemma: For z = (z1 , · · · , zn ) ∈ Cn , we write |z| = maxi |zi |. For fixed z with |z| < 1, set g(t) = f (tz) for t ∈ C. Then g(t) is a holomorphic function in the disk ∆ = |t| ≤ |z|−1 and the first h coefficients of its Taylor series at 0 vanish. Therefore, g(t)/th is still holomorphic in the ∆. By maximal principal, in the disk |g(t)/th | ≤ M/|z|−h = M |z|h . Setting t = 1, we obtain (8.27). Now we proceed to prove Siegel’s theorem. Proof: Polynomials in n variables of degree ≤ k form a vector space whose dimension is k+n (k + 1)(k + 2) · · · (k + n) = n! n which is a polynomial of degree n viewed as a function in k. let f1 , · · · , fn+1 be n + 1 meromorphic functions on a compact complex manifold of dimension n, we want to show there exists a polynomial F of n + 1 variables such 96
that F (f1 , · · · , fn+1 ) = 0 We choose three neighborhoods of an arbitrary x ∈ X, denoted by Wx ⊂ Vx ⊂ Ux . Ux is chosen so that Pi,x fi = Qi,x for i = 1, · · · , n, n + 1 where Pi,x , Qi,x are holomorphic in Ux and are relatively prime at each point y ∈ Ux . we note the existence of Ux follows the previous Lemma 1. Vx is chosen so that Vx ⊂ Ux and Vx has a local coordinate system (z1 , · · · , zn ) with |z| := maxi |zi | < 1. Wx given by |z| < 1/2. For two distinct points x and y, since Pi,x and Qi,x (resp. Pi,y and Qi,y )are relatively prime, it follows Qi,x = Qi,y ϕ(i) x,y (i)
where ϕx,y are holomorphic and nowhere vanishing in Ux ∩ Uy . From the system of neighborhoods Wx , we choose finite cover [ X= Wξj 1≤j≤r
where ξj denote centers of the r open balls. The finite covering is possible due to the compactness of X. We set ϕx,y =
n+1 Y
ϕ(i) x,y , C = maxx,y maxVx ∩Vy |ϕx,y |
i=1
We note ϕx,y is holomorphic and bounded in Vx ∩ Vy since its closure is contained in Ux ∩ Uy . Moreover C ≥ 1 since ϕx,y · ϕy,x = 1. For a polynomial F (t1 , · · · , tn+1 ) of degree k, we set F (f1 , · · · , fn+1 ) =
Rx Qkx
in Vx , where Qx =
n+1 Y
Qi,x
i=1
and Rx =
(i) (ϕx,y )k
· Ry in Vx ∩ Vy .
To construct such a polynomial F , we first show that for any given h, F can be chosen so that F 6≡ 0 and Rξj ∈ mhξi for all the r points in the finite covering of X. These conditions can be written out as the set of relations (Ds Rξj (ξj )) = 0 97
for j = 1, 2, · · · , r, where Ds is the partial derivative of order s < h. Hence they are linear relations on the coefficients of F ; the number of these relations equals r n+h−1 . n If we choose the degree k of F such that n+k+1 n+h−1 >r (8.28) n+1 n then there exists a non-zero polynomial F such that F (f1 , · · · , fn+1 ) = 0. By Schwarz’s lemma, for this choice of F , the functions Rξj will be small in the neighborhoods Wξj . Indeed, writing M = maxξj maxx∈Vξj |Rξj (x)| we have, for x ∈ Wξj M (8.29) 2h This circumstance implies that M = 0, that is the vanishing condition F (f1 , · · · , fn+1 ) = 0 holds for sufficiently large k and h. Indeed, suppose that the maximum value M is attained at a point x0 ∈ Vη , then x0 ∈ Wξν ⊂ Vξν for some point ξν . Hence, |Rξj (x)| ≤
M = |Rη (x0 )| = |Rξν (x0 )| · |ϕξν ,η (x0 )|k Now if k and h are such that (8.28) and thus (8.29) hold, we have M≤
M k C 2h
Now if we further require that k and h are chosen so that C k /2h < 1 also holds, we get M = 0. But this is possible: since C ≥ 1, we write C = 2λ for some λ ≥ 0, then we only need to take h ≥ λk, as long as (8.28) is satisfied. With similar arguments it can be shown that if the transcendence degree of M(X) equals k and f1 , · · · , fk are algebraically independent meromorphic functions on X, then the degree of the irreducible relation F (f, f1 , · · · , fk ) = 0 satisfied by an arbitrary meromorphic function f is upper bounded. Therefore M(X), in addition to being of finite transcendence degree, is also finitely generated.
98
Chapter 9 Hermitian and K¨ ahler Manifolds 9.1
Hermitian manifolds
Let M be a 2n-dimensional smooth manifold. An almost complex structure on M is an isomorphism J : T M → T M of the tangent bundle T M such that the J 2 = −1. In other words, it consists of a family of isomorphisms Jx : Tx M → Tx M such that Jx2 = −1 and the assignment x 7→ Jx is smooth. A smooth manifold equipped with almost complex structure is called an almost complex manifold. Conversely, if a smooth M admits an almost complex structure, it must must have even dimension. Indeed, if dim(M ) = n = 2k + 1, then Pn (λ) = det(J − λI) is a polynomial of odd degree in variable λ, which must have a real root λ0 , there is an eigenvector v ∈ T M such that J(v) = λ0 v and J 2 (v) = λ20 v 6= −v since λ0 is real. This is a contradiction to J = −1. Let X be an n-dimensional complex manifold, the complex structure gives rise to a J-operator. Let√(zj ),1 ≤ j ≤ n be a system of local holomorphic coordinates on X with zj = xj + −1yj , then in terms of the usual basis { ∂x∂ j , ∂y∂ j }, 1 ≤ j ≤ n, the J-operator is defined as J( ∂x∂ j ) = ∂y∂ j and J( ∂y∂ j ) = − ∂x∂ j . Then we can extend J by complex linearity into the complexified tangent bundle TXC . √ For any x ∈ X, TxC splits into direct sum of the (± −1)-eigenspaces, denoted by Tx1,0 and Tx0,1 respectively. Indeed, suppose η ∈ TxC is an eigenvector, then we have J(η) = λη for some non-zero η ∈ C. From J 2 (η) = λ2 η = −η, we obtain that ¯ η , hence η, η¯ belong to the T 1,0 λ = ±i. By taking the conjugate, we have J(¯ η ) = λ¯ x and Tx0,1 respectively. Writing TX1,0 = ∪x∈X Tx1,0 (reps. TX0,1 ) as the subbundle of TXC , we have the bundle decomposition TXC = TX1,0 ⊕ TX0,1 . In terms of local coordinates, √ a basis for Tx1,0 is ∂z∂ j = 21 ( ∂x∂ j − −1 ∂y∂ j ), 1 ≤ j ≤ n; Similarly, a basis for Tx0,1 is √ ∂ 1 ∂ = ( + −1 ∂y∂ j ), 1 ≤ j ≤ n. ∂ z¯j 2 ∂xj 99
Definition(Hermitian metric): A Hermitian metric g on a complex manifold X is a J-invariant Riemannian metric on the underlying smooth manifold X. In other words, for real tangent vectors u and v, g satisfies g(u, v) = g(J(u), J(v)). A complex manifold X equipped with a Hermitian metric g is called a Hermitian manifold, denoted by (X, g). Extending g by complex-bilinearity to TXC , we obtain a symmetric C-bilinear form g(., .). The J-invariant condition is equivalent to that g(u, v) = 0 for any u, v of the same type. Indeed, for u,v ∈ TxC , if u v ∈ Tx1,0 , then J(u) = iu and J(v) = iv, therefore g(u, v) = g(J(u), J(v)) = g(iu, iv) = i2 · g(u, v) = −g(u, v), which implies g(u, v) = 0; Similarly, if both u and v are in Tx0,1 , then J(u) = −iu, J(v) = iv, therefore we also have g(u, v) = g(J(u), J(v)) = g(−iu, −iv) = g(u, v), and g(u, v) = 0. In the local holomorphic coordinates (zj ), 1 ≤ j ≤ n, we have an n by n matrix G = (gi¯j ) given by gi¯j = g( ∂z∂ i , ∂∂z¯j ). We note that it can be decomposed into two parts: 1 ∂ ∂ ∂ ∂ 1 ∂ ∂ 1 ∂ 1 ∂ gi¯j = g( , ) + g( , ) + [ g( , ) − g( , )] · i = Aij + Bij · i 4 ∂xi ∂xj 4 ∂yi ∂yj 4 ∂xi ∂yj 4 ∂xj ∂yi (9.1) ∂ ∂ where Aij and Bij are real-valued. And clearly, for gj¯i = g( ∂ z¯i , ∂zj ) = Aji + Bji · i, we have Aji = Aij while Bji = −Bij . Hence gj¯i = gi¯j . Therefore, the matrix G is Hermitian symmetric and the Hermitian metric g can be written as ! X
i
j
j
i
{gi¯j dz ⊗d¯ z +gj¯i dz ⊗d¯ z}=
1≤i
X
i
j
z j } = 2Re {gi¯j dz ⊗d¯ z +gi¯j dz i ⊗ d¯
X
i
gi¯j dz ⊗ d¯ z
1≤i
1≤i
(9.2) Meanwhile, associated with g, there is a real tensor ! Im
X
gi¯j dz i ⊗ d¯ zj
(9.3)
1≤i
P P P i j i j i j Since Im g dz ⊗ d¯ z +Im g d¯ z ⊗ dz = Im g dz ⊗ d¯ z = ¯ ¯ ¯ i j j i i j 1≤i
j
Definition(K¨ahler metric): A Hermitian manifold (X, g) is said to be K¨ahler if and only if the types of complexified tangent vectors are preserved under parallel transport. For a Hermitian manifold (X, g), it is K¨ahler is equivalent to the following conditions: 1. for any parallel real vector field η along a smooth curve γ, J(η) is also parallel; 2. ∇J ≡ 0, i.e., the almost complex structure is J is parallel; 3. ∇ω ≡ 0, i.e., the fundamental form ω is parallel; 4. dω ≡ 0, i.e., the fundamental form ω is closed; 5. locally, in terms of a holomorphic coordinates (zj ), 1 ≤ j ≤ n, there exists a 2 potential function ϕ such that gi¯j = ∂z∂j ∂ϕz¯j ; 6. at every point P ∈ X, there exists a complex geodesic coordinates (zi ) in the sense that the Hermitian metirc g is replaced by the Hermitian matrix (gi¯j ) satisfying gi¯j (P ) = δij and dgi¯j = 0. Proof: Original definition ⇒ (1): Suppose η is a parallel real vector field along a smooth curve γ := {γ(t) : − < t < }. Let η = η 1,0 + η 0,1 be the unique decomposition of the vector field into components of type (1, 0) and (0, 1). let ϕ and ϕ0 be the parallel transport of η 1,0 (0) and η 0,1 (0) along γ. By definition, ϕ and ϕ0 are vector fields of type (1, 0) and (0, 1) respectively and by the uniqueness of decomposition of tangent vectors into type (1, 0) and (0, 1), it follows that ϕ ≡ η 1,0 0 0,1 and γ. Hence η 1,0 and η 0,1 are both parallel along γ. Since J(η) = √ ϕ1,0≡ η√ along 0,1 −1η − −1η , we conclude that J(η) is also parallel along γ. (1) ⇒ original definition: Let η 1,0 (0) be a tangent vector of type (1, 0) at γ(0) and write ϕ for its parallel transport along γ. Let ϕ = ϕ1,0 +ϕ0,1 be the decomposition into √ 1,0 components of types (1, 0) and (0, 1). By condition (2), we have J(ϕ) = −1ϕ − √ −1ϕ0,1 is also parallel. It follows that both ϕ1,0 and ϕ0,1 are parallel along γ. Since ϕ0,1 (0), we conclude that ϕ0,1 vanishes identically on γ so that the parallel transport of η 1,0 (0) remains to be of type (1, 0) along γ. (1) ⇒ (2): Let P ∈ X and v be any real tangent vector at P . Choose the curve γ such that γ(0) = P and v is tangent to γ. Let η be a parallel vector field along γ. From condition (1), we have that J(η) is also parallel, i.e., ∇v (J(η)) = 0. By product rule, ∇v J(η) = J∇v η + (∇v J)(η). Since ∇v η = 0, we have (∇v J)(η) =. Since η(0) is arbitrary at P , it follows ∇JP = 0, whence ∇J ≡ 0 for P is also arbitrary. (2) ⇒ (1): With the same notation, we have, for a parallel real vector field η along γ that ∇v (η) = 0. By assumption ∇v J = 0, hence by product rule ∇v (J(η)) = J∇v η + (∇v J)(η) = 0, i.e., J(η) is also parallel along γ. 101
P (2) ⇔ (3): Now we consider the tensor G = 1≤i
then hij (0) = δij and
∂hij (0) ∂xk
Γkij
= 0. In particular we have Γkij (0) = 0, where
1 X kl := h 2 l
∂hil ∂hil ∂hij + − ∂xj ∂xi ∂xl
P Back to our proof, We now write the Riemannian metric as ij hij dxi ⊗ dxj . We choose the normal geodesic coordinates at an arbitrary point P ∈ X, the hypothesis (3) implies dhij (P ) = 0 and it follows dω = 0. √ ¯ = ω. We fix an open set U on (4) ⇒ (5): It suffices to solve the equation −1∂ ∂ϕ X which is biholomorphic to an Euclidean polydisk. Since ω is d-closed, by Poincar´e lemma, there exists a real 1-form η on U such that dη = ω. Decompose η into 102
¯ 1,0 + components of types (1, 0) and (0, 1) as η = η 1,0 + η 0,1 . Then we have (∂ + ∂)(η 0,1 η ) = ω. Since ω itself is of type (1, 1), there are no (2, 0)-forms or (0, 2)-forms ¯ 0,1 = 0 and ω = ∂η ¯ 1,0 + ∂η 0,1 . By bar∂in this identity, hence we have ∂η 1,0 = ∂η 0,1 ¯ Poincar´e lemma, ∂η = 0 implies that there exists a local smooth function φ such 0,1 ¯ that ∂ϕ = η . Since η 1,0 = η 0,1 , we have η 1,0 = ∂ ϕ, ¯ therefore, ¯ 1,0 + ∂η 0,1 = ∂(∂ ¯ φ) ¯ + ∂(∂φ) ¯ = ∂ ∂(φ ¯ − φ) ¯ ω = ∂η √ √ ¯ it solves the equation − −1∂ ∂ϕ ¯ = ω. Now let ϕ = − −1(φ − φ), √ ¯ then (5) ⇒ (4): If there is a such a potential function ϕ such that ω = −1∂ ∂ϕ, √ √ √ √ ¯ − −1∂ ∂¯2 ϕ = 0 ¯ = −1(∂ + ∂(∂ ¯ ∂ϕ) ¯ = −1∂ 2 ∂ϕ dω = −1d(∂ ∂ϕ) (5) ⇒ (6): Let P ∈ X be an arbitrary point. By making a unitary change of coordinates, we may assume that coordinates (w1 , · · · , wn ) have √ thePholomorphic been chosen so that writing ω = −1 hi¯j dwi ∧ dw¯ j , we have hi¯j (P ) = δij . Now we make a holomorphic change of coordinates from (w1 , · · · , wn ) to (z 1 , · · · , z n ) in the following way: X cijk z j z k wi = z i + j,k
such that cijk = cikj . Suppose ω =
√
−1
P
gi¯j dz i ∧ d¯ z j with gi¯j (P ) = δij , then we have
∂gi¯j ∂hi¯j ∂hi¯j (P ) = (P ) + cjki + cjik = (P ) + 2cjik ∂zk ∂zk ∂zk This equation vanishes precisely if cjik = − 12
∂hi¯j , ∂zk
which is possible if and only if
∂hi¯j ∂hk¯j (P ) = (P ) ∂zk ∂zi which is equivalent to dω(P ) = 0. (6) ⇒ (3): Using complex geodesic coordinates (zj ) at P ∈ X, the √ connection Γ P from i gi¯j dw ∧ dw¯ j , of the underlying Riemannian manifold vanishes at P . Writing ω = −1 we obtain ∇ω(P ) = 0.
9.2
Hermitian Connections and Curvature Forms
Let X be a complex manifold and E be a holomorphic vector bundle over X. Let h be a Hermitian metric on E, i.e., a collection of Hermitian inner products on the fibre EP for any P ∈ X, varying smoothly with P . A connection, which essentially is 103
a way of differentiating smooth sections of the vector bundle over open sets is defined in the following way: Definition(Hermitian connection): A connection D on a complex vector bundle (E, h) is a map such that, for any smooth section s of V over an open set U and any tangent vector fields ξ on an open set U , Dξ s is also a smooth section of V over U , C-linear in both ξ and s, satisfying the product rule Dξ (f · s) = ξ(f ) · s + f + f · Dξ s
(9.4)
for any f ∈ C ∞ (U ). A connection D on V is said to be compatible with complex structure (or, simple a complex connection) if and only if for any local section si and any tangent vector η = η 1,0 + η 0,1 over the domain of si , we have Dη0,1 si = 0. It is said to be compatible with the Hermitian metric (or, simply a metric connection) if and only the following holds: for any open set U , for any real tangent vector on v on U and for any smooth sections s and t over U , we have v(t, s) = (Dv t, s) + (t, Dv s) and for any complexified tangent vector ξ it follows, ξ(t, s) = (Dξ t, s) + (t, Dξ¯s) A connection D on V is a Hermitian connection as we defined above with respect to a Hermitian metric h is equivalent to being a complex, metric connection. Indeed, the requirement that D be complex is consistent with product rule since the transition functions for V are holomorphic. Let U be a coordinate open set on X with holomorphic local coordinates (zi ) such that V is holomorphically trivial over U . Let {eα } be P a holomorphic for V |U and we write s = sα eα for a smooth section of V over Pbasis ∂ U . Let η = ηi ∂zi be a smooth vector field of type (1, 0) over U . By the product rule, to define a complex connection D of V , it suffices to define Di eα = Γγiα , where Γγiα is the Riemann-Christoffel symbol and Di stands for D∂/∂zi . For this purpose, we apply the additional condition that D be metric so we have ∂i (eα , eβ ) = (Di eα , eβ ) + (eα , D¯i eβ )
(9.5)
where (eα , eβ ) is the Hermitian inner product, which will be written as hαβ¯. The matrix (hαβ¯)αβ , denoted by H is Hermitian symmetric and positive definite. By the assumption that D is complex, the term D¯i eβ in (9.5) is zero, hence (9.5) becomes hγ β¯Γγiα = ∂i hαβ¯ ¯
¯
(9.6)
i.e., Γγiα = hγ β · ∂i hαβ¯ where (hαβ ) is the conjugate inverse of the matrix (hαβ¯). Now it is clear that the Riemann-Christoffel symbol (Γγiα ) defined by (9.5) gives rise to a unique complex metric connection D on (E, h). 104
Let (X, g) be a Hermitian manifold. The restriction of the Hermitian metric to TX1,0 defines a Hermitian metric on TX1,0 , which can be identified with the holomorphic tangent bundle TX . By conjugation D extends to a connection on TXC = TX1,0 ⊕ TX0,1 . On the other hand, since by definition, g is a Riemannian metric on the underlying smooth manifold X, there is a Riemannian connection ∇ on (X, g), which extends to the complexified tangent bundle TXC . Definition(Riemannian Connection): A connection ∇ on a Riemannian manifold (M, g) is a Riemannian connection if and only if 1. it is compatible with the metric g, i.e.,for any vector fields X and Y and η ∈ TxC ∂η (X, Y ) = (∇η X, Y ) + (X, ∇η Y ) 2. and is torsion free, i.e., T (X, Y ) := ∇X Y, ∇Y X − [X, Y ] = 0 Compare the two connections D and ∇, we have the following theorem. Theorem: The Hermitian connection D agrees with the Riemannian connection ∇ if and only if (X, g) is K¨ahler. In other words, a Hermitian manifold (X, g) is K¨ahler if and only if the Hermitian connection D √ is torsion free. P Proof: Write G = gi¯j dz i ⊗ d¯ z j = S + −1A as we did before, where S and A are ¯ is the underlying Riemannian metric of the smooth manifold real tensors. 2S = D+ G ¯ = Dη¯G, from DG ∼ X. Since for any complexified tangent vector η, we have Dη G = 0, ∼ we deduce that DS = 0. In other words, D is compatible with the Riemannian metric given by 2S. Then by the uniqueness of the Riemannian connection, if follows that D∼ = ∇ if and only of D is torsion free. We note
∂ ∂ , =0 ∂zi ∂ z¯j
And since D is complex connection, we have Di ( ∂∂z¯j ) = D¯i ( ∂z∂ j ) = 0, if then follows T (η, ξ) = 0 if η and ξ are of opposite types. On the other hand, we have ∂ ∂ ∂ ∂ ∂ ∂ T , = Di ( ) − Dj ( −) − , ∂zi ∂zj ∂zj ∂zi ∂zi ∂zj =
X
(Γkij − Γkji )
k
∂ ∂zk
By taking the conjugate, we obtain a similar formula for T 105
∂ , ∂ ∂ z¯i ∂ z¯j
.
It follows that the Hermitian connection D is torsion free if and only the subscriptions of the Riemann-Christoffel symbols we defined earlier for D commute,.i.e Γkij = Γkji for all i, j and k. For any P ∈ X, we choose holomorphic coordinates (zi ) such that gi¯j (P ) = δij , it then follows ¯∂gj ¯ l Γkij = g kl = ∂i gj k¯ ∂zi Thus, the Hermitian connection D is torsion free if and only if ∂i gj k¯ (P ) = ∂j gik¯ (P ), i.e. dgi¯j (P ) = 0, for any P ∈ X, which is equivalent to (X, g) being K¨ahler. With respect to a system of local holomorphic coordinates (zi )1≤i≤n over an open set U and a local holomorphic basis {eα }1≤ℵ≤r of V |U over U , we express the Hermitian connection of a Hermitian holomorphic vector bundle (E, h) of rank r in terms of Riemann-Christoffel symbols in the following form Di eα = Γγiα eγ We define an End(V )-valued 1-form Γ = Γγiα eγ ⊗ eα ⊗ dz i . We call Γ the connection 1-form on (E, h) over U . For Riemannian connections, Γ depends on the choice of the local holomorphic coordinates (zi ) and the holomorphic basis {eα } on V |U . We can identify Γ with a row vector with r entries as Γ1iα eα ⊗ dz i , · · · , Γriα eα ⊗ dz i which, will also be denoted by Γ. Let e = (e1 , · · · , er )T , then we can express the Hermitian connection D as De = Γ ⊗ e Let f be a row vectors of r entries, each of which is a smooth function on U , then, by product rule, we have D(f · e) = df ⊗ e + f · De
Now we will give a definition of curvature of the Hermitian holomorphic vector bundle (E, h) of rank r with respect to the Hermitian connection D. Let ν be an E-valued p-form on U . Denote by Dν the covariant derivatives of ν. For an E-valued 1-form ϕ ⊗ e, we have D(ϕ ⊗ e) = dϕ ⊗ e − ϕ ∧ De For the flat connection d, we have d2 (f · e) = 0. For (E, h), we have D2 (f · e) = D(df ⊗ e + f · De) d2 f ⊗ e − (df ∧ Γ) ⊗ e + (df ∧ Γ) × e + f · D2 e f · D2 e = f · dΓ ⊗ e − f · (Γ ∧ Γ) ⊗ e 106
where Γ ∧ Γ is matrix multiplication on forms as Γ ∧ Γ = (Γγiα eα ⊗ eγ ⊗ dz i ) ∧ Γσjβ eσ ⊗ eσ ⊗ dz j ) = (Γµiα Γσjµ − Γµjα Γσiµ )eα ⊗ eσ ⊗ (dz i ∧ dz j ) We define the curvature of (E, h) to be the End(E)-valued 2-form as √ θ = −1(dΓ − Γ ∧ Γ)
(9.7)
In terms of the frame {e∗i ⊗ ej } for E ∗ ⊗ E, θ can be interpreted as a r-by-r matrix of 2-forms so that we have √ D2 (f · e) = − −1f · Θ ⊗ e (9.8) In this sense, our definition of Θ is independent pf the choice of the local coordinates (si ). Theorem: The curvature Θ associated with a holomorphic vector bundle is a (1, 1)form. To prove the theorem, we need a lemma as follows. Lemma: For an arbitrary point P ∈ X and a holomorphic coordinate system on neighborhood U of P , over which the vector bundlePE is trivial, we can find a basis {eα } of E|U such that for the Hermitian metric h = hαβ¯eα ⊗ e¯β such that hαβ¯(P ) = δαβ and dhαβ¯(P ) = 0. Proof of the lemma: Choose the holomorphic coordinates (zi ) such that zi (P ) = 0. Let {e0α } be a holomorphic basis of E|U with dual basis {e∗α 0 } such that the matrix (hαβ¯) representing h with respect to {e0α } satisfies h0αβ¯(P ) = δαβ . Now we change the coordinates from {e0α } to {eα } with dual basis {e∗α } by e∗α = e∗α 0 + clγ α zl eγ then we have hαβ¯(P ) = δαβ and ∂l hαβ¯(P ) = ∂l h0αβ¯(P ) + cαl β 0 By setting cαl β to −∂l hαβ¯ (P ), we obtain that dhαβ¯ = 0.
This particular basis {eα } of E|U is called a special holomorphic basis at P . And we note that the choice is independent of the choice of holomorphic coordinates (zi ). Recall that the existence of complex geodesic coordinates (zi ) such that for a Hermitian metric g, gi¯j (P ) = δij and dgi¯j = 0 is equivalent to (X, g) being K¨ahler. The choice of a special holomorphic basis locally for a holomorphic vector bundle over a Hermitian manifold is always possible, even if it is not K¨ahler. Now we proceed to the proof of the theorem. 107
Let P be an arbitrary point on X and {eα } be special holomorphic basis at P . A special local for the fibre EP such that for any v ∈ E, we P frame (vα ) is so chosen ¯ γ write v = vα eα . Recall that Γiα = (∂i hαβ¯)hγ β while by our choice of basis, we have dhαβ¯, i.e., (∂i hαβ¯) = 0 for any i, thus, by the expression of Γ we have seen before, √ this term will vanish, and we are left with −1dΓ. Recall that Γ = (∂H)H −1 for the matrix H = (hαβ¯)αβ . Then, in terms of (vα ), we obtain Θ=
√
−1dΓ =
√ √ ¯ −1d((∂H)H −1 ) = −1∂∂H
This shows that the curvature Θ is of type (1, 1), which can be written as √ √ Θ = −1Θβα eα ⊗ eβ = −1 Θβαi¯j eα ⊗ eβ dz i ∧ d¯ zj
108
(9.9)
(9.10)
Chapter 10 Kodaira Embedding Theorem 10.1
Kodaira-Nakano Vanishing Theorem
In this section, we denote by X a compact complex K¨ahler manifold. Definition: A line bundle L√ on X is positive if there exists a metric on L with a −1 Θ is a positive (1, 1)-form; L is negative if the dual curvature form Θ such that 2π bundle L∗ is positive. The following theorem gives another characterization of the positivity of a line bundle. 2 Theorem: Let ω be a real closed (1, 1)-form on X such that [ω] = c√1 (L) ∈ HdR (X). −1 There exists a metric connection on L with curvature form Θ = 2π ω. The line bundle L is positive if and only if its first Chern class c1 (L) can be represented by a 2 (X). positive form in HdR
To proof the theorem, we first introduce a lemma. Lemma: Let η be a (p, q)-form on a complex K¨ahler manifold such that η is d-,∂- or ¯ ¯ ∂-exact. Then √ η = ∂ ∂γ for some (p − 1, q − 1)-form γ. If p = q and η is real, then we may take −1γ also to be real. Proof: Let Gd be the Green’s operator associated to the Laplacian ∆d and similarly for G∂ and G∂¯. Recall the identity that 1 ∆d = ∆∂ = ∆∂¯ 2 It follows 2Gd = G∂ = G∂¯ 109
¯ ∗ ,∂ ∗ and ∂¯∗ all commute with the Green’s operator. and then all the operators d,∂, ∂,d ¯ Now since η is d-, ∂- or ∂-exact, its Hermitian projection under any of the above ¯ we have Laplacian is zero. By Hodge decomposition for ∂, η = ∂¯∂¯∗ G∂¯η But ∂¯∗ G∂¯η has pure type (p, q − 1) so we have ∂(∂¯∗ G∂¯η) = ±∂¯∗ G∂¯(∂η) = 0 Since the harmonic space for ∂ is the same as the harmonic space for ∂¯ and hence is orthogonal to the range of ∂¯∗ , we deduce by the Hodge decomposition for ∂ that ¯ ∗ G ¯η) ∂¯∗ G∂¯η = ∂∂ ∗ G∂ (∂) ∂ By commuting the various operators we obtain ¯ ∗ ∂¯∗ G2¯η) η = ±∂∂(∂ ∂ which implies the lemma. Proof to the theorem: Let |s|2 be a metric on L with curvature form Θ and let ϕ : L|U → U × C be a trivialization of L over an open neighborhood U of X. Let s : U → L|U ∼ = U × C, z 7→ (z, sU (z)) be a section of L over U , then |s|2 = h(z)|sU |2 , where h(z) is √a real-valued positive function. Suppose that L is positive, then we can −1 Θ is a positive (1, 1)-form. Now the curvature form is given by assume that 2π ¯ Θ = −∂ ∂logh(z) which is a real, closed (1, 1)-form. The first Chern class is given by √ −1 2 c1 (L) = [ Θ ∈ HdR (X)] 2π Now let |s0 |2 be another metric on L with curvature form Θ0 . Then |s0 |2 /|s|2 = eρ for some real smooth function ρ on X, and from the local formula 0 ρ ρ ¯ ¯ ¯ ¯ +Θ Θ0 = ∂ ∂logh (z) = −∂ ∂loge h(z) = −∂ ∂(loge + logh(z) = −∂ ∂ρ
we have the identity
√
√ −1 −1 0 Θ = Θ 2π 2π
√
−1 Conversely, suppose that 2π ϕ is a real, closed (1, 1)-form representing c1 (L) in ( HdR M ). If we can solve the equation
¯ +ϕ Θ = ∂ ∂ρ 110
(10.1)
for a real smooth function ρ, then the metric eρ |s|2 on L will have curvature form ϕ. The solvability of (10.1) follows from our first lemma. This theorem gives an equivalent definition of positive line bundle. Definition: Let L be a holomorphic line bundle on X and let c1 (L) be its first Chern class. Then L is said to be positive if there exists a real, closed (1, 1)-form η such that η ∈ c1 (L) and η is positive. Let Ωp,q (X) be the space of (p, q)-forms on X and let ω be the K¨ahler form on X. We define the operator L : Ωp,q (X) −→ Ωp+1,q+1 (X) by ξ 7→ ξ ∧ ω and let Λ : Ωp,q −→ Ωp−1,q−1 be the adjoint operator. Then we have the following identities √ √ ¯ = − −1∂ ∗ , [Λ, ∂] = −1∂¯∗ , [L, Λ] = p + 1 − n [Λ, ∂] where n = dim(X). Let E be a vector bundle on X and let Ωp,q (E) be the space of E-valued (p, q)-forms. Consider the operator ∆ = ∂¯∂¯∗ + ∂¯∗ ∂¯ : Ωp,q (E) → Ωp,q (E) We let Hp,q (E) be the kernel of the above operator. It is the space of E-valued harmonic forms. By harmonic theory we know that H q (X, Ωp (E)) ∼ = Hp,q (E) Now let D√= D0 + D00 with D00 = ∂¯ be the metric connection on E. By the identity [Λ, ∂¯ = − −1∂ ∗ , we obtain √ ¯ = − −1D0∗ [Λ, ∂] The following vanishing theorem will be fundamental in the proof of Kodaira embedding theorem in the next section. Theorem(Kodaira-Nakano): Let L be a positive line bundle on a complex K¨ahler manifold X of dimension n. Then we have H q (X, Ωp (L)) = 0 for any p + q > n. Proof: By harmonic theory we know that H q (X, Ωp (L)) ∼ = Hp,q(L) . We want to show there are no L-valued harmonic forms of degree larger than n = dim(X). 111
√
−1 By hypothesis, there exists a metric on L such that ω = 2π Θ, where Θ is the curvature form associated to the metric and ω is the K¨ahler form on X. Now let η ∈ Hp,q (L) be a harmonic form. By interpreting the curvature operator Θη = Θ ∧ η alternatively as √2π L(η), and as D2 η, where D is the metric connection on L, we −1 have, for any harmonic form η ∈ Hp,q (L)
¯ 0 + D0 ∂¯ Θ = D2 = ∂D ¯ = 0, it follows and from ∂η ¯ 0η Θη = ∂D Therefore
√ √ ¯ 0 η, η) 2 −1(ΛΘη, η) = 2 −1(Λ∂D √ √ −1 0∗ 0 ¯ − D )D η, η = (D0∗ D0 η, η) = (D0 η, D0 η) ≥ 0 = 2 −1 (∂Λ 2 ¯ 0 η, η) = (ΛD0 η, ∂) ¯ ∗ η) = 0. Similarly, since (∂ΛD √ √ ¯ 2 −1(ΘΛη, η) = 2 −1(D0 ∂Λη, η) √ √ −1 0∗ 0 ¯ = 2 −1 D (Λ∂ + D )η, η = −(D0 D0∗ η, η) = −(D0∗ η, D0∗ η) ≤ 0 2 Combining, we have But Θ =
√2π L, −1
√ 2 −1([Λ, Θ]η, η) ≥ 0
and so
√ 2 −1([Λ, θ]η, η) = 4π([Λ, L]η, η) = 4π(n − p − q)kηk ≥ 0 Thus if p + q > n, then η = 0.
10.2
Kodaira Embedding Theorem
In this section, we will be concerned with the situation when a compact complex manifold is an algebraic variety, i.e., when it can be embedded into a projective space PN . Let X be a compact complex manifold and L → X be a holomorphic line bundle. To any subspace E of the vector space H 0 (X, O(L)), there is an associated linear system |E| = {(s)}s∈E ⊂ Div(X) of divisors on X. Since X is compact, (s) = (s0 ) only if s = λ · s0 for some non-zero constant λC; thus |E| is parametrized by the projective space P(E). 112
Suppose in addition, our linear system has no base point,i.e., there is no such a point P ∈ X such that all s ∈ E vanish at P . Then for each P ∈ X, the set of sections s ∈ E vanishing at P forms a hyperplane H˜P ⊂ E, or, equivalently, the set of divisors D ⊂ |E| containing P forms a hyperplane HP in P(E), and we can define a map ι : X → P(E)∗ by sending P ∈ X to HP ∈ P(E)∗ . The map is described in terms of a basis s0 , · · · , sN for E. Let si,α = ϕ∗α (si ) ∈ O(U ) for any trivialization ϕα of L over an open set U ∈ X. As we have seen before, the coordinate [s0,α (P ), · · · , sN,α (P )] ∈ PN is independent of the trivialization chosen. Hence we can remove the subscript α and denote this point by [s0 (P ), · · · , sN (P )]. In terms of the identification P(E)∗ ≡ PN corresponding to the choice of basis s0 , · · · , sN , the map ι is give by ι(P ) = [s0 (P ), · · · , sN (P )] which is holomorphic. Now let H be the hyperplane bundle on PN . The pullback bundle ι∗P (H) on X is (∗) given by the divisor (si ), i.e., any section s = ai si ∈ E is P L = ι (H). Moreover, the pullback of the section ai Zi of H on PN ,i.e., we have E = ι∗ (H 0 (PN , O(H))) ⊂ H 0 (X, O(L)) Thus ι : X → PN determines both the line bundle L and the subspace E ⊂ H 0 (X, O(L)). Now we want to answer the question that given L → X a holomorphic line bundle, when will the corresponding map ιL : X → PN be an embedding. First, in order for ιL to be well defined, the linear system |L| shall have no base points, namely, for any x ∈ X, the restriction map rx H 0 (X, O(L)) −→ Lx must be surjective. Assume this, the map ιL is an embedding if
1. ιL is one-to-one. Clearly, this is the case if and only if for all x and y in X, there exists a section s ∈ H 0 (X, O(L)) vanishing at x but not at y, or, equivalently, the restriction map rx,y H 0 (X, O(L)) −→ Lx ⊗ Ly (10.2) is surjective for x 6= y ∈ X. We note when (10.2) is true, the base-point-free condition for the linear system |L| will also hold true. 2. ιL has non-zero differential everywhere. If ϕα is a trivialization of L near x, then this is the case if and only if for all v ∗ ∈ Tx∗ (X), there exists s ∈ H 0 (X, O(L)) with sα (x) = 0 and dsα (x) = v ∗ where sα = ϕ∗α (s). More intrinsically, this requirement can be expressed in the following way: 113
Let Jx ∈ O be the sheaf of holomorphic functions on X vanishing at x. For any section s ∈ Jx (L) near x and any trivializations ϕα and ϕβ of L in a neighborhood U of x, writing sα = ϕ∗α (s), sβ = ϕ∗β (s) with sα = gαβ sβ , we have, at x d(sα ) = d(sβ ) · gαβ + dgαβ · sβ = d(sβ ) · gαβ Thus we have a well-defined sheaf map 0
dx : Jx (L) −→ Tx∗ ⊗ Lx So this condition can be stated as requiring that the map d
0
x Tx∗ ⊗ Lx H 0 (X, Jx (L)) −→
(10.3)
be surjective for all x ∈ X. We note that (10.3) is the limiting case of (10.2) as y → x. The result we are aiming for is stated in the following theorem. Theorem(Kodaira Embedding): Let X be a compact complex manifold and L → X be a positive line bundle. Then there exists k0 such that for k ≥ k0 , the map ιLk : X → PN is well-defined and is an embedding of X. Proof: Let L → X be a positive line bundle on the compact complex manifold X, we prove that there exists k0 such that 1. The restriction map rx,y : H 0 (X, O(Lk )) → Lkx ⊕ Lky is surjective for all x 6= y ∈ X, k ≥ k0 , and 2. The differential map dx : H 0 (X, Jx (Lk )) → Tx0∗ ⊗ Lkx is surjective for all x ∈ X, k ≥ k0 . Before we proceed to the proof, we first introduce the concept of blowing up and a lemma that will be used in the proof. Let (zi )1≤i≤n be the Euclidean coordinates in a disk ∆ and l = [l1 , · · · , ln ] be the ˜ be a submanifold of ∆×Pn−1 corresponding homogeneous coordinates on Pn−1 . Let ∆ given by the quadratic relations ˜ = {(z, l) : zi · lj = zj · li for all i, j} ∆ 114
(10.4)
If we consider points l ∈ Pn−1 as lines in Cn , then writing these equations as z ∧ l we see that this is just the incidence correspondence defined as {(z, l) : z ∈ l}. ˜ maps onto ∆ via the projection on the first factor π : (z, l) 7→ z; intuitively, this ∆ is an isomorphism away from the origin in ∆, and π −1 (0) is is just the projective ˜ consists of disjoint lines through the origin in ∆. Now, ∆ ˜ space of lines in ∆ while ∆ together with its projection π to ∆ is called the blow-up of ∆ at the origin. Now for a compact complex manifold X of dimension n, and any x ∈ X, let z : U → ∆ be a coordinate polydisc centered at x. The restriction of the projection map ˜ − E → U − {x} π:∆ ˜ and a neighborgives an isomorphism between a neighborhood of E = π −1 (x) in ∆ ˜ x of X at x to be the complex manifold hood of x in X; We define the blow-up X ˜ x := X − {x} ∪π ∆ ˜ X ˜ together with the natural projection map π : X ˜ → X. Again by replacing ∆ with ∆ the projection ˜ x − {π −1 (x)} → X − {x} π:X ˜ x is called exceptional divisor of the is an isomorphism; the pre-image π −1 (x) in X blow-up. Lemma: KX˜ = π ∗ KX + (n − 1)E. Proof: We first prove the lemma in the case when X has a non-trivial meromorphic n-form ω. In terms of local coordinates (zi ) in a neighborhood U of x, we write ω(z) =
f (z) 1 · dz ∧ · · · ∧ dz n g(z)
Then U˜ = π −1 (U ) = {(z, l) ∈ U × U × Pn−1 : zi · lj = zj · li } and we set U˜i = (li 6= 0) ⊂ U˜ . In this way we obtain an open covering of the neighborhood U˜ of E and in each open set U˜i we have local coordinates z(i)j defined as lj zj z(i)j = = li zi ˜ x → X is given in U˜i by for i 6= j. The map π : X (z(i)j , · · · , z(i)n , · · · , z(i)n ) 7→ (z(i)1 · zi , · · · , zi , · · · , z(i)n · zi ) and so π∗ω = π∗
f (n−1) · zi dz(i)1 ∧ · · · ∧ dz(i)n g 115
Thus we see that in a neighborhood of E = π −1 (x0 ), the divisor (π ∗ ω) is given by π ∗ (ω) + (n − 1)E. Since clearly (π ∗ ω) = π ∗ (ω) away from E, we have KX˜ = [(π ∗ ω)] = π ∗ KX + (n − 1)E as desired. To prove the lemma in the general cases (without the assumption that X has a nontrivial meromorphic n-form), we let U = {U0 , Uα }α be an open coordinate covering of X with x ∈ U0 , x 6∈ Uα and all the sets Uα having non-empty intersection with U0 lying in one coordinate patch with coordinates z1 , · · · , z)n. Let U˜ = {U˜α = π −1 Uα , U˜i = π −1 U0 ∩ (li 6= 0)} ˜ we compute the transition functions {gij , giα , gαβ } be a corresponding covering for X; for KX˜ in terms of the coordinates z(i)j on U˜i and wi,α = π ∗ wi,α on U˜α where {wi,α }i are coordinates on Uα in X. First we have, in U˜i ∩ U˜2 , z(2)1 = z(1)−1 2 z2 = z(1)2 · z1 z(2)i = z(1)i · z(1)−1 2 for i 6= 1, 2, and so the Jacobian matrix for the change of coordinates is
J12
=
−z(1)−2 2 z1 0
0 z(1)2 0 .. .
0 ··· 0 ··· 0 ···
0 0 0 z(1)−1 2
−z(1)j · z(1)−2 0 · · ·
0 .. .
In general, gij = detJij = z(1)−n+1 j Similarly, in U˜α ∩ U˜1 , w1,α = z1 and wi,α = z1 · z(1)i , J1α
1 .. .
0 ···
= z(1)i 0 · · · .. . 116
0
···
z1 · · ·
0
0
and in general (n−1)
giα = zi
0 0 Also we have gαβ = π ∗ gαβ , where gαβ are the transition functions for KX with respect to coordinates wi,α in Uα , wi,β in Uβ .
Now E is given in U˜i by (zi ), in U˜α by (1); so the transition functions for [E] over U˜ are zi hij = = z(i)−1 j zj hiα = zi hαβ = 1 Thus the transition functions for the bundle KX˜ ⊗ [E]−n+1 are fij = z(i)−n+1 · z(i)jn−1 = 1 j fiα =i zin−1 ·i zi−n+1 = 1 fαβ = π ∗ gαβ and we see that KX˜ − (n − 1)E is just the pullback via π of the bundle on X given by transition functions e0α = 1, eαβ = gαβ i.e., KX˜ − (n − 1)E = π ∗ KX . Now we can proceed to the proof of the embedding theorem. ˜ → X be the blow-up of X at both x and y, Ex = π −1 (x) and To prove (1), let π : X −1 Ey = π (y) the exceptional divisors of the blow-up. We denote the pullback π ∗ L by ˜ Consider the pullback on the sections L. ˜ O ˜ (Lk )) π∗ : H 0 (X, OX (Lk )) → H 0 (X, X ˜ k , the section Lk given by σ over X − {x, y} extends by For any global section σ ˜ of L Hartogs’ theorem to a global section σ ∈ H 0 (X, O(Lk )) and so we see that π ∗ is an ˜ k is trivial along Ex and Ey , i.e., isomorphism. Furthermore, by defintion, L ˜ k )|Ex = Ex × Exk , (L ˜ k )|Ey = Ey × Eyk (L ˜ k )) ≡ Lk ⊕ Lk , where E is the divisor Ex + Ey . so that H 0 (E, OE (L x y Let rE the restriction mp to E, then the following diagram E ˜ O ˜ (L ˜ k )) −−r− ˜ k )) H 0 (X, → H 0 (E, OE (L X x
rx,y
H 0 (X, O(Lk )) −−−→ 117
Lkx ⊕ Lky
(10.5)
commutes. Thus to prove (1) for x and y, it suffices ti show the restriction rE is ˜ surjective. For this purpose, we construct the following exact sequence on X rE ˜ k − E) → O ˜ (L ˜ k ) −→ ˜k) → 0 0 → OX˜ (L OE (L X
(10.6)
∗ is positive on X. And we can choose k2 such that Choose k1 such that Lk1 + KX k ˜ ˜ L − nE is positive on X for k ≥ k2 . By the previous lemma we have
˜ X + (n − 1)E KX˜ = K ˜ X = π ∗ KX . Thus for k ≥ k0 = k1 + k2 , where K ∗ ˜ k − E) = Ωn˜ (L ˜ k − E + K ∗˜ ) = Ωn˜ (L ˜ k1 + K ˜X ˜ k0 − nE) OX˜ (L ) ⊗ ( L X X X 0
˜ k − nE has a positive definite curvature form on X; ˜ for k 0 ≥ k2 . Now by hypothesis, L ∗ k1 k1 ∗ ˜ ˜ L +KX has a positive definite curvature form X, whose L +KX becomes a positive ˜ This implies that the line bundle (L ˜ k0 − nE) + (L ˜ k1 + K ˜∗ ) semi-definite form on X. X ˜ By Kodaira vanishing theorem, it follows is positive definite on X. ˜ k − E) = H 1 X, ˜ Ωn˜ ((L ˜ k0 − nE) + (L ˜ O ˜ (L ˜ k1 + K ˜ ∗ )) = 0 for k ≥ k0 H 1 X, X X X Hence the restriction map ˜ k )) → H 0 (E, OE (L ˜ k )) ˜ O ˜ (L rE : H 0 (X, X is surjective for k ≤ k0 and so (1) is proved for x and y. ˜ → X be the blow-up of X at a single point x with exceptional To prove (2), let π : X −1 divisor E = π (x). Again, the pullback ˜ O ˜ (Lk )) π ∗ : H 0 (X, OX (Lk )) → H 0 (X, X is an isomorphism. Further, if σ ∈ H 0 (X, OX (Lk )), then σ(x) = 0 if and only if σ ˜ = π ∗ (σ) vanishes on E; thus π ∗ restricted to the sheaf J of sections of holomorphic functions on X vanishing at x is an isomorphism ˜ O ˜ (L ˜ k − E)) π ∗ : H 0 (X, Jx (Lk )) → H 0 (X, X With the identification ˜ k − E)) = Lk ⊗ H 0 (E, OE (−E) ∼ H 0 (E, OE (L = Lkx ⊗ Tx0∗ x the diagram E ˜ O ˜ (L ˜ k − E)) −−r− ˜ k − E)) H 0 (X, → H 0 (E, OE (L X x
∗
π
H 0 (X, Jx (Lk ))
d
x −−− →
118
Tx0∗ ⊗ Lkx
(10.7)
˜ w have the commutes. It suffices to prove that rE is surjective for k 0. On X following exact sequence rE ˜ k − 2E) → O ˜ (L ˜ k − E) −→ ˜ k − E) → 0 0 → OX˜ (L OE (L X
(10.8)
˜ k0 − Again, we choose k1 such that Lk1 + KX ∗ is positive on X and k2 such that L ˜ for k 0 ≥ k2 . Then, for k ≥ k0 = k1 + k2 (n + 1)E is positive on X ∗ ˜ k − 2E) = Ωn˜ (L ˜ k1 + K ˜X ˜ k0 − (n + 1)E) OX˜ (L ) ⊗ ( L X with k 0 ≥ k0 ; hence rE is surjective on global sections and (2) is proved for an arbitrary fixed x. It remains to find one value of k0 such that (1) and (2) hold for all choices of x and y and for all k ≥ k0 . But if the map ιLk has been defined at x, y, with ιLk (x) 6= ιLk (y), the same will be true for x0 near x and y 0 near y. Similarly, if our ιLk is smooth at x, it will also be smooth at x0 for any x0 near x and separate points near x. Since X is compact, the result follows. Kodaira embedding theorem can be restated as following: A compact complex manifold X is an algebraic variety, i.e., is embeddable in project space if and only if it has a closed, positive (1, 1)-form ω whose cohomology class [ω] is rational.
Proof: If [ω] ∈ H 2 (X, Q), then for some k, [kω] ∈ H 2 (X, Z). Now we consider the exact sequence exp i ∗ 0 → Z −→ OX −→ OX →0 Then in the corresponding exact sequence for cohomology class is f
i
∗ ∗ · · · → H 1 (X, OX ) −→ H 2 (X, Z) −→ → ···
i∗ ([kω]) = 0. So [kω] ∈ Ker(i∗ ) = Image(f ). Since H 1 (M, O)X ∗ ) = Pic(X), there exists a holomorphic line bundle L → X with c1 (L) = [kω] and the line bundle is positive. The result follows Kodaira embedding theorem. Conversely, if X ⊂ PN is embedded in a projective space, then the restriction of the Fubini-Study metric of PN on X gives a K¨ahler metric on X whose associated K¨ahler form is an integral, positive and closed (1, 1)-form on X. A metric whose (1, 1)-form is rational is called a Hodge metric. It follows that any compact Hodge manifold X admits an embedding in a projective space. The dimension of the first secant variety on X is given by exp{dim(Sec(X))} = 2n + 1
(10.9)
Hence dim(Sec(X)) ≤ 2n + 1. By iterated projection we can embed any compact Hodge manifold of dimension n into P2n+1 . 119
In the next section, we will apply Kodaira embedding theorem to a compact Riemann surface to show that any compact Riemann surface is embeddable.
10.3
Embedding of Riemann surfaces
In this section, X will be used to denote a compact Riemann surface. We will prove the following theorem. Theorem: Any compact Riemann surfaces can be embedded in P3 as a projective curve. To prove the theorem, we need a lemma. Lemma: Let 2i ω be a positive (1, 1)-form on a compact Riemann surface X. There exists a holomorphic line bundle L → X and a Hermitian metric whose curvature form is a positive (1, 1)-form. Proof: Let U = {Uα } be a covering of X by coordinate charts with each Uα simply connected. On Uα , we write i zα ω = fα dzα ∧ d¯ 2 where fα is positive and smooth. We can solve the ∂¯ equation twice on Uα to obtain a 2 ¯ α on Uα . hα may be local potential function hα such that fα = ∂∂z∂hαz¯ and thus ω = ∂ ∂h 2 complex-valued, but since the operator ∂∂∂¯ is the complexification of the real operator and fα is real-valued, the imaginary part of hα is harmonic and can be safely ignored. Thus, we may assume that hα is real-valued. Now we consider the function hα := hα − hβ defined on Uαβ := Uα ∩ Uβ . These functions are harmonic so we can find holomorphic functions Gαβ on Uαβ with Re(Gαβ ) = hαβ . Let gαβ := eGαβ , we will modify gαβ to obtain the transition functions for the holomorphic line bundle. −1 To meed the cocycle condition, i.e. gαβ · gβγ · gαγ = 1, we have, in terms of Gαβ ,
(δG)αβγ := Gαβ + Gβγ − Gαγ = 2πi · m for some integer m. Since the real part of Gαβ is hαβ and the function hαβ form a splitting cocycle, it follows that (δG)αβγ is pure imaginary. On the other hand, (δG)αβγ is also holomorphic, it is a pure imaginary constant. Multiplying by i, we see that i(δG) ∈ Z 2 (U, R). Now by Leray’s theorem and the simply connectivity, we have H 2 (U, R) ∼ = H 2 (X, R) ∼ =R 120
Let [i(δG)] be the cohomology class of i(δG), since H 2 (U, Z) ∼ = Z, there exists some non-zero cocycle η in Z 2 (U, Z) ⊂ Z 2 (U, R). Multiplying by a real non-zero constant c, we can arrange that [i(δG)] = [η], i.e., there is a 1-cochain Kαβ in C 1 (U, R) with i · c(δG)αβγ − (δK)αβγ=ηαβγ ∈ Z. Replacing η by −η if necessary, we can assume c > 0. Let ˜ αβ := 2π(cGαβ + iKαβ ) G (10.10) ˜
and let g˜αβ := eGαβ , then the functions are holomorphic, satisfying the cocycle re˜ αβ ) = lation. Therefore, g˜αβ determine a holomorphic line bundle L. Also Re(G ˜ α := 2πc · hα , and define the Hermitian metric on L locally by 2πc(hα − hβ ). Define h ˜ e2hα , the curvature form Θ of this metric is ¯ h ˜ α ) = (4πc)∂ ∂h ¯ α = (4πc)ω Θ = ∂ ∂(2
(10.11)
as desired. We note that the definition is independent of the choice of coordinates. Indeed, for two charts Uα and Uβ , we have ˜
e2hα e2h˜ β
˜
= e4πc(hα −hβ ) = e2Re(Gαβ ) = |˜ gαβ |2
Proof to the theorem: On a Riemann surface, the tangent bundle of type (1, 0) is a holomorphic line bundle. A holomorphic line bundle always admits a Hermitian metric. We write a Hermitian metric on this holomorphic bundle as eϕ . This metric gives us an associated volume form dV on X which can be written locally as i z dV = eϕ dx ∧ dy = eϕ dz ∧ d¯ 2 √
which is positive, and is a candidate for the positive (1, 1)-form 2−1 ω in the previous lemma. Now applying the lemma, we obtain a Hermitian metric for the holomorphic line bundle whose curvature form Θ = (4πc)ω with c > 0. Therefore √ √ −1 −1 Θ = 4c( ω) 2π 2 is a positive (1, 1)-form. By Kodaira embedding theorem, the compact complex Riemann surface X is embeddable. Moreover, recall that a Hodge manifold of dimension n can be embedded into P2n+1 by iterated projections, it follows, X can be embedded into P3 .
121
Selected Bibliography Including Cited Works [1] K. Chandrasekharan. Elliptic functions, volume 281. 1985. [2] O. Forster. Lectures on Riemann surfaces. Springer Verlag, 1981. [3] P. Griffiths and J. Harris. Principles of algebraic geometry, volume 1994. Wiley, 1978. [4] R. Miranda. Algebraic curves and Riemann surfaces, volume 5. Amer Mathematical Society, 1995. [5] N. Mok. Metric rigidity theorems on Hermitian locally symmetric manifolds, volume 6. World Scientific Pub Co Inc, 1989. [6] R. Narasimhan. Compact Riemann Surfaces. Birkhauser, 1992. [7] G. Springer. Introduction to Riemann surfaces. Chelsea Pub Co, 2002. [8] H.P.F. Swinnerton-Dyer. Analytic theory of abelian varieties. Cambridge University Press, 1974.
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