International Journal of Mathematics, Vol. 12, No. 2 (2001) 159–201 c World Scientific Publishing Company

DIMENSIONAL REDUCTION, SL(2, C)-EQUIVARIANT BUNDLES AND STABLE HOLOMORPHIC CHAINS

´ ´ LUIS ALVAREZ-C ONSUL Department of Mathematics, University of Illinois Urbana, IL 61801, USA E-mail : [email protected] OSCAR GARC´IA-PRADA Departamento de Matem´ aticas, Universidad Aut´ onoma de Madrid, 28049 Madrid, Spain E-mail : [email protected]

Received 1 May 2000 In this paper we study gauge theory on SL(2, C)-equivariant bundles over X × P1 , where X is a compact K¨ ahler manifold, P1 is the complex projective line, and the action of SL(2, C) is trivial on X and standard on P1 . We first classify these bundles, showing that they are in correspondence with objects on X — that we call holomorphic chains — consisting of a finite number of holomorphic bundles Ei and morphisms Ei → Ei−1 . We then prove a Hitchin–Kobayashi correspondence relating the existence of solutions to certain natural gauge-theoretic equations and an appropriate notion of stability for an equivariant bundle and the corresponding chain. A central tool in this paper is a dimensional reduction procedure which allow us to go from X × P1 to X. Keywords: Equivariant bundle, stability, Hermitian–Einstein equations, vortex equations, holomorphic chains, Hitchin–Kobayashi correspondence, dimensional reduction.

0. Introduction Many fundamental partial differential equations in mathematical physics and gauge theory appear as a dimensional reduction of the instanton equations on a four-dimensional Riemannian manifold. Examples of these include the Bogomolny monopole equations in three dimensions [4], as well as Hitchin self-duality equations on a Riemann surface [24]. Another important example in dimension 2 is given by the abelian vortex equations [26]. These are equations originally considered on the Euclidean plane R2 , which first appeared in the Ginzburg-Landau model for superconductivity. A mathematical study of these equations was done by Taubes [26, 36] who proved existence of solutions and showed that they are a dimensional reduction of the SU(2)-instanton equations on R2 times the 2-dimensional sphere. The vortex solutions (the so-called vortices) correspond precisely to the instantons which are invariant under the action on the sphere of the group of 3-dimensional 159

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rotations. This followed previous work of Witten [39], who had obtained vortices on the hyperbolic plane in a similar way. In [18, 19] Garc´ıa–Prada showed that the vortex equations on a compact Riemann surface, and in fact the generalization to higher dimensional K¨ ahler manifolds studied by Bradlow [5], can also be obtained via dimensional reduction. In the higher dimensional case the usual 4-dimensional instanton equations do not make sense of course, and one has to consider a dimensional reduction of another set of fundamental equations on a K¨ ahler manifold — the Hermitian–Einstein equations — which in complex dimension 2 coincide with the instanton equations. It turns out that the vortex equations on a line bundle over a K¨ahler manifold X correspond to the SU(2)-invariant solutions of the Hermitian–Einstein equations on a certain SU(2)-equivariant rank-2 vector bundle over X × P1 , where SU(2) acts trivially on X and in the standard way on P1 . The vortex solutions are determined by a pair consisting of a holomorphic line bundle and a holomorphic section of this bundle, i.e. an effective divisor on the K¨ahler manifold. Using this pair one constructs a holomorphic rank-2 vector bundle on X × P1 . Now, by the theorem of Donaldson, Uhlenbeck and Yau [14, 15, 38], the existence of a Hermitian–Einstein metric on this bundle is equivalent to the stability of the bundle in the sense of Mumford– Takemoto — this is the well-known Hitchin–Kobayashi correspondence. The construction above was extended in [20] to pairs (E, φ) consisting of a holomorphic vector bundle of arbitrary rank and a holomorphic section. In [6], Bradlow studied the non-abelian vortex equations for such pairs and proved a theorem establishing a correspondence between the existence of solutions and certain stability condition for the pair. Bradlow’s stability condition depends on a real parameter, which is encoded in the K¨ahler metric on X × P1 . The equivariant bundle on X × P1 corresponding to a pair (E, φ) on X is given as an extension that has the pull-back of E as a subbundle and the pull-back of O(2) — the cotangent bundle of P1 — as the quotient. It was realized in [20] that one could enlarge the picture by considering more general equivariant bundles on X × P1 given by extensions in which the quotient is now twisted with the pull-back of another holomorphic bundle on X. These correspond to triples (E1 , E2 , φ) consisting of two holomorphic vector bundles over X and a morphism φ : E2 −→ E1 . Of course, a pair is just a triple in which one of the bundles is the trivial homolorphic line bundle. The study of invariant solutions to the Hermitian–Einstein equation on the equivariant bundle over X × P1 led to some new equations on the triple (E1 , E2 , φ), known as the coupled vortex equations [20]. A systematic study of these equations and a proof of a Hitchin–Kobayashi-type correspondence, establishing the equivalence between the existence of solutions and a certain stability condition for the triple, was done by Bradlow and Garc´ıa–Prada in [12]. In this paper we deal with the general case. We first classify equivariant bundles on X × P1 and then study the dimensional reduction of the corresponding gauge theory equations and stability criteria. It turns out that a general equivariant bundle

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on X × P1 is in correspondence with an object on X consisting of a finite number of holomorphic vector bundles Ei over X and morphisms φi : Ei → Ei−1 . We call such an object a holomorphic chain. The Hermitian–Einstein equations for invariant solutions on the equivariant bundle lead to vortex-type equations on the chain, which generalize the triple coupled vortex equations. One realizes immediately that these equations should naturally involve as many real parameters as morphisms in the chain. However, the Hermitian–Einstein equations can only account for one parameter which, as mentioned above, is encoded in the K¨ ahler polarization on 1 X × P necessary to define the Hermite–Einstein condition. But every equivariant bundle on X × P1 is naturally filtered — the extensions appearing in the case of pairs and triples are just one-step filtrations — and, to make up for the rest of the parameters, one must consider a certain deformation of the Hermitian–Einstein equation. This deformation, which involves as many parameters as steps in the filtration, has been studied for one step filtrations by Bradlow and Garc´ıa–Prada [9] and Daskalopoulos, Uhlenbeck and Wentworth [17]. In the case of triples the parameter in the deformation of the Hermitian–Einstein equations can be traded with the change of the K¨ ahler metric on X × P1 , but this cannot be done in the general case. The paper is organized as follows. After some basic definitions and preliminaries on equivariant bundles and equivariant sheaves, in Sec. 1, we classify SL(2, C)equivariant holomorphic bundles over X × P1 . The group SL(2,C) (the complexification of SU(2)) acts trivially on X and in the usual way on P1 via the identification P1 = SL(2, C)/P , where P is the subgroup of lower triangular matrices. Exploiting the basic representation theory of P , we give a description of SL(2, C)-equivariant bundles on X × P1 in terms of two equivalent categories. On the one hand they can be described as certain kind of filtrations on X × P1 and, on the other, as holomorphic chains on X. In Sec. 2, we undertake the study of the notion of stability, the deformed Hermitian–Einstein equations and the Hitchin– Kobayashi correspondence for a filtration. Since this is of independent interest we do it on any compact K¨ ahler manifold. This generalizes previous work on stable holomorphic extensions [9], and has been extended by Mundet i Riera [30] to more general K¨ ahler fibrations. The filtrations on X × P1 that we have to deal with are in fact SL(2, C)-equivariant, and hence to finish the section we deal with an equivariant version of the Hitchin–Kobayashi correspondence for filtrations. In Sec. 3, we turn our attention to the category of holomorphic chains on a compact K¨ ahler manifold X. We introduce natural vortex equations, a notion of stability, and prove a Hitchin–Kobayashi correspondence for chains. Finally, in Sec. 4, we show how the chain vortex equations and the stability of a chain on X are a dimensional reduction of the modified Hermitian–Einstein equations and the stability condition, respectively, on the corresponding equivariant filtration on X × P1 . One important application of the theory of stable pairs and triples is to the study of moduli spaces of representations of the fundamental group of a surface

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in a complex or real non-compact Lie group [35]. Hitchin [24] showed that these moduli spaces have an algebro-geometric interpretation as moduli spaces of Higgs bundles (analogous to the interpretation of unitary representations as stable vector bundles, via the theorem of Narasimhan ans Seshadri [32]), and furthermore, he introduced the idea of applying Morse-Bott theory to these moduli spaces. To carry out this programme it is crucial to have information about the critical submanifolds of the Morse function, the so-called variations of Hodge structure. The critical submanifolds have (in some cases of Higgs bundles of low rank) been identified with moduli spaces of vortex pairs or triples and this has led to topological information about the moduli spaces of representations (see Hitchin [24], Gothen [22, 23]). It turns out that the variations of Hodge structure found as critical points in higher rank Higgs bundle moduli spaces correspond to moduli spaces of stable chains for certain special values of the stability parameters. We show this for the complex special group in Sec. 3.3. From this point of view it is then important to construct the moduli space of stable chains and study their topology and geometry. One expects similar flip transformations to those studied by Thaddeus [37] and Bradlow, Daskalopoulos and Wentworth [8] for the case of pairs to relate the moduli spaces for different values of the parameters. We hope to come back to this in the future. The coupled vortex equations for triples have also been related by Bradlow, Glazebrook and Kamber [11, 12] to the Hermitian–Einstein equations on vector bundles over certain non-trivial P1 -fibrations over X. It seems clear that a similar picture could be applied to general holomorphic chains. The study of holomorphic chains is the first step in the study of the more general vector bundle quivers considered by King [27], whose approach has been very influential in our work. Some of the more general quivers are also related to dimensional reduction by considering an appropriate equivariant manifold [2]. Abelian vortices have become more relevant recently because of their relation to Seiberg–Witten theory [16, 40]. It turns out that the Seiberg–Witten equations, which are generally defined on a four-dimensional Riemannian manifold, reduce to the vortex equations when the manifold is K¨ ahler (see e.g. [21]). On the other hand, the more classical vortices on a Riemann surface do appear in the study of Floer–Seiberg–Witten theory on the product of the Riemann surface by the circle [31]. It is conceivable that the more general vortex equations that we are studying in this paper may be of future significance in this connection. 1. Equivariant Bundles Let X be a complex manifold. Consider the product manifold X × P1 , where P1 is the complex projective line. The group SL(2, C) acts on X × P1 by the trivial action on X and the standard action on P1 via the identification P1 = SL(2, C)/P , where P is the parabolic subgroup of lower triangular matrices of SL(2, C). This section is primarily devoted to the study of SL(2, C)-equivariant holomorphic vector bundles and sheaves on X × P1 . We give a description of these objects in terms

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of two equivalent categories: SL(2, C)-equivariant holomorphic and sheaf filtrations on X × P1 , and holomorphic and sheaf chains on X (Theorem 1.1). The method is one relating SL(2, C)-equivariant sheaves on X × P1 to P -equivariant sheaves on X, through the induction process, and through the weight decomposition of P equivariant sheaves on X with respect to the complex torus C∗ ⊂ P , both explained in Sec. 1.1. In order to fix notation, we first review in Sec. 1.1 some basic definitions and preliminaries about equivariant vector bundles and sheaves on general manifolds M with a Lie group action. 1.1. Basic definitions and preliminaries Given a real Lie group K, a smooth K-manifold is a smooth manifold M together with an action of K on M by diffeomorphisms. A K-equivariant smooth complex vector bundle on a K-manifold M is a smooth complex vector bundle F on M with a smooth lifting of the K-action on M to F . If we consider instead a complex Lie group G acting by biholomorphisms on a complex manifold M , we say that M is a complex G-manifold; analogously a G-equivariant holomorphic vector bundle on a complex G-manifold is given by a holomorphic lifting of the G-action on M to a holomorphic vector bundle F . On a complex G-manifold M there is also a natural notion of coherent G-equivariant (analytic) sheaf (cf. e.g. [1]), which for complex algebraic varieties acted by an algebraic complex Lie group G reduces to the familiar notion of coherent G-linearized algebraic sheaf. Let X be a complex manifold and let X × P1 have the SL(2, C)-action defined at the beginning of the section. The classification of SL(2, C)-equivariant holomorphic vector bundles and coherent sheaves on X × P1 given in Sec. 1.3 depends heavily on two processes, inverse to each other, usually called induction and restriction, which we briefly describe in the sequel. First of all, any SL(2, C)-equivariant holomorphic vector bundle F on X × P1 defines by restriction a P -equivariant holomorphic vector bundle E on the slice X × P/P ∼ = X. On the other hand, if E is a P equivariant holomorphic vector bundle on X, then F = SL(2, C) ×P E is an SL(2, C)-equivariant holomorphic vector bundle on X × P1 = X × SL(2, C)/P . Here F = SL(2, C) ×P E is the quotient of SL(2, C) × E by the action of P on both factors, by p · (g, e) = (gp−1 , p · e), and the action of g ∈ SL(2, C) on SL(2, C) ×P E is given by g ′ · [g, e] = [g ′ g, e]. This construction generalizes to coherent equivariant sheaves. Indeed there is a one-to-one correspondence between isomorphism classes of coherent SL(2, C)-equivariant sheaves S on X × P1 and isomorphism classes of coherent P -equivariant sheaves V on X, which puts in bijection the locally free sheaves. The correspondence is as follows. Any coherent SL(2, C)-equivariant sheaf F on X ×P1 defines by restriction a coherent P -equivariant sheaf on X ×P/P ∼ = X. Conversely, if E is a coherent P -equivariant sheaf on X, then it induces the coherent SL(2, C)-equivariant sheaf F = SL(2, C) ×P E on X × P1 . The first step to describe coherent P -equivariant sheaves on X is to consider the decomposition into a direct sum of its weight subsheaves with respect to the

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action of the complex torus C∗ ⊂ P . This is given by the following proposition, whose proof is immediate. Proposition 1.1. Let Mλ , for λ ∈ Z, denote the irreducible C∗ -module with weight λ (i.e. z · v = z λ v for z ∈ C∗ , v ∈ Mλ ), as well as its associated C∗ -equivariant constant sheaf. Then every coherent C∗ -equivariant sheaf E on a compact complex manifold X, considered as a complex C∗ -manifold with trivial C∗ -action, is isomorL phic to a finite direct sum λ Eλ ⊗ Mλ , where Eλ are coherent sheaves with trivial C∗ -action. If E is locally free then the sheaves Eλ are also locally free. 1.2. P -modules Our study of coherent SL(2, C)-equivariant sheaves on X × P1 or, equivalently, of coherent P -equivariant sheaves on X will be simplified by considering first P modules, i.e. finite-dimensional complex representations of P . As we shall see, P modules admit two equivalent descriptions, both of them with a counterpart in the case of equivariant sheaves. Let us recall that the Lie algebra sl(2, C) is generated by the three matrices ! ! ! 0 0 0 1 1 0 , , X− = , X+ = H= 1 0 0 0 0 −1 with the commutation relations [H, X + ] = 2X + , [H, X − ] = −2X −, [X + , X − ] = H , and the Lie algebra p ⊂ sl(2, C) of the subgroup P of lower triangular matrices is generated by H and X − . Since P is not semisimple, its representations are not always semisimple. Let W be a representation of P . This is given by a representation of p in which the Cartan subalgebra h = spanC {H} of sl(2, C) acts diagonally with integer eigenvalues. Hence the action of the complex torus C∗ ∼ = exp(h) ⊂ P , is given by a weight decomposition M Wλ ⊗ Mλ . (1.1) W = λ∈∆(W )

Here Mλ , for each λ ∈ Z, is the (one-dimensional) irreducible representation of C∗ with weight λ (cf. Proposition 1.1), Wλ = HomC∗ (Mλ , W ) are the complex vector spaces of C∗ -equivariant endomorphisms (the multiplicity subspaces), and ∆(W ) = {λ ∈ Z|Wλ 6= 0} is the set of weights of W . The rest of the P -module structure of W is determined by the action of X − . Since [H, X − ] = −2X −, the action of X − on Wλ ⊗ Mλ is obtained from a linear map ϕλ : Wλ −→ Wλ−2 , and the trivial X − -action on Mλ . Thus a P -module is exactly specified by a diagram of linear maps ϕ4

ϕ2

ϕ−2

ϕ0

· · · −→ W2 −→ W0 −→ W−2 −→ · · · ϕ5

ϕ3

ϕ1

ϕ−1

· · · −→ W3 −→ W1 −→ W−1 −→ · · · .

(1.2)

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Obviously, every P -module is the direct sum of an even part (i.e. Wλ = 0 for λ odd) and an odd part (i.e. Wλ = 0 for λ even). Furthermore, every indecomposable P -module must be described by a sequence ∆(W ) = {m− , m− +2, . . . , m+ −2, m+ } of consecutive odd or even integers (the set of weights), and for such a set, by a sequence Wm− , Wm− +2 , . . . , Wm+ of non-zero finite dimensional vector spaces, and linear maps ϕλ : Wλ → Wλ−2 between consecutive non-zero vector spaces. After twisting by M−m− , one can always get a representation V = W ⊗ M−m− whose C∗ -action is described by V =

m M i=0

Vi ⊗ M2i ,

Vi := Wm− +2i ,

(1.3)

so its set of weights is shifted to ∆(V ) = {0, 2, . . . , 2m}, m := (m+ − m− )/2, and whose X − -action is given by linear maps φi := ϕm− +2i : Vi → Vi−1 , for 1 ≤ i ≤ m. This can be represented schematically by the diagram φm

φm−1

φ1

Vm −→ Vm−1 −→ · · · −→ V0 ,

(1.4)

which gives our first description of a P -module mentioned above. It is obvious that the P -module W is indecomposable if and only if the diagram (1.4) is indecomposable in the obvious sense. The second description is given in terms of flags of P -modules. Consider an indecomposable P -module V which, after twisting by some M−m− , gets the structure given by (1.3) and (1.4). This P -module admits obviously the following flag of P -submodules: V(≤•) : 0 ⊂ V(≤0) ⊂ V(≤1) ⊂ · · · ⊂ V(≤m) , V(≤i) /V(≤i−1) ∼ = Vi ⊗ M2i ,

0 ≤ i ≤ m,

(1.5)

where V(≤i) =

M j≤i

Vj ⊗ M2j .

Observe that each component gri := V(≤i) /V(≤i−1) of the associated grading gr• = ⊕i gri of V(≤•) is a semisimple representation of P , and each of the P -submodule V(≤i) in V is the extension of P -modules 0 −→ V(≤i−1) −→ V(≤i) −→ gri −→ 0 determined by φi . Summing up, we have seen that any indecomposable P -module, after twisting by some irreducible representation of C∗ , admits a description in terms of a chain (1.4) of linear maps between its multiplicity spaces, and a flag of P -modules (1.5) whose P -submodule V(≤i) (0 ≤ i ≤ m) is an extension of a semisimple P -module gri by the preceding nested P -submodule V(≤i−1) .

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1.3. Equivariant bundles in terms of chains and equivariant filtrations The study of coherent SL(2, C)-equivariant sheaves on X × P1 is now straightforward, and yields analogues of the preceding pictures (1.4) and (1.5). To begin with, let T be a coherent SL(2, C)-equivariant sheaf on X × P1 . As explained in Sec. 1.1, its restriction S = T |X to X ∼ = X × P/P ⊂ X × P1 , defines a P -equivariant sheaf. The SL(2, C)-equivariant sheaf T is recovered from S by induction: T ∼ = SL(2, C) ×P S .

(1.6)

From Proposition 1.1, S admits a weight space decomposition as a C∗ -equivariant sheaf, M Sλ ⊗ Mλ S= λ∈∆(S)

where ∆(S) ⊂ Z is the set of eigenvalues for the C∗ -action on S. As for P -modules, the rest of the P -equivariant structure is determined by the action of X − , which yields morphisms ϕλ : Sλ → Sλ−2 since [H, X − ] = −2X −. This leads to a diagram similar to (1.2), ϕ2

ϕ4

ϕ−2

ϕ0

· · · −→ S2 −→ S0 −→ S−2 −→ · · · ϕ5

ϕ3

ϕ−1

ϕ1

(1.7)

· · · −→ S3 −→ S1 −→ S−1 −→ · · · . Every indecomposable P -equivariant sheaf S has a sequence ∆(S) = {m− , m− + 2, . . . , m+ − 2, m+ } of consecutive odd or even integers (cf. the argument above for indecomposable P -modules). After twisting by M−m− , so as to get the equivariant sheaf E = S ⊗ M−m− , the set of weights is shifted to ∆(E) = {0, 2, . . . , 2m}. Hence the H-action is given by E=

m M i=0

Ei ⊗ M2i ,

Ei = Sm− +2i

(1.8)

and the X − -action is determined by the diagram φm

φm−1

φ1

C : Em −→ Em−1 −→ · · · −→ E0 .

(1.9)

This gives our first description of coherent SL(2, C)-equivariant sheaves: Definition 1.1. A sheaf chain on X is a pair C = (E, φ), where E = (E0 , E1 , . . . , Em ) is an (m + 1)-tuple of coherent sheaves on X, and φ = (φ1 , . . . , φm ) is an m-tuple of homomorphisms φi ∈ Hom(Ei , Ei−1 ) (1 ≤ i ≤ m). This is represented by the diagram (1.9). We say that C is a holomorphic chain if the sheaves Ei are locally free. Note that sheaf chains form an abelian category, where the morphisms are defined in the obvious way (cf. Definition 3.2 below).

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We can see the effect of the twist E = T ⊗ M−m− in terms of sheaves on X ×P1 . Indeed, the coherent P -equivariant sheaf E gives rise to a coherent SL(2, C)equivariant sheaf F on X × P1 by induction, F = SL(2, C) ×P E ,

(1.10)

and the twisting E = S ⊗ M−m− of sheaves on X is equivalent to the twisting F = T ⊗ q ∗ O(−m− )

(1.11)

of SL(2, C)-equivariant sheaves on X × P1 (here q : X × P1 → P1 is projection). This is simply because for any integer λ, O(λ) = SL(2, C) ×P Mλ . We now consider our second description, in terms of SL(2, C)-equivariant sheaf filtrations on X × P1 . Given an indecomposable coherent SL(2, C)-equivariant sheaf on X × P1 , we have seen that, after a twisting (1.11), its restriction E = F|X to X is given by (1.8) and (1.9). This coherent P -equivariant sheaf admits the following filtration of coherent P -equivariant subsheaves: E (≤•) : 0 ֒→ E(≤0) ֒→ E(≤1) ֒→ · · · ֒→ E(≤m) = E , E(≤i) /E(≤i−1) ∼ = Ei ⊗ M2i ,

0 ≤ i ≤ m,

(1.12)

where E(≤i) =

M j≤i

Ej ⊗ M2j .

We now recall that our original SL(2, C)-equivariant sheaf F on X ×P1 was induced by E (cf. (1.10)), and hence we obtain a filtration of SL(2, C)-equivariant subsheaves of F : F : 0 ֒→ F0 ֒→ F1 ֒→ · · · ֒→ Fm = F , Fi /Fi−1 ∼ = Ei ⊠ O(2i) ,

0 ≤ i ≤ m,

(1.13)

where the subsheaves Fi are induced by E(≤i) : Fi := SL(2, C) ×P E(≤i) . Here we have used the notation E ′ ⊠ E ′′ = p∗ E ′ ⊗ q ∗ E ′′ for the external tensor product of a sheaf E ′ on X and a sheaf E ′′ on P1 , where p : X × P1 → X and q : X × P1 → P1 are the corresponding projections. Remark 1.1. Note that, as in the case of the P -submodules of the flag V in the last section, each component gri := Fi /Fi−1 of the associated grading gr• = ⊕i gri of F is semisimple with respect to the action of P , and each of the P -submodule F≤i in F is the extension of coherent equivariant SL(2, C)-sheaves 0 −→ Fi−1 −→ Fi −→ gri −→ 0 determined by φi ∈ Hom(Ei , Ei−1 ).

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Definition 1.2. (a) A sheaf filtration is a finite sequence of coherent subsheaves F : 0 ֒→ F0 ֒→ F1 ֒→ · · · ֒→ Fm = F .

(1.14)

We say that F is a holomorphic filtration if the sheaves Fi are locally free. (b) An SL(2, C)-equivariant sheaf filtration on X × P1 is a sheaf filtration F as in (1.14), together with a structure of SL(2, C)-equivariant coherent sheaf on F which induces a structure of an SL(2, C)-equivariant coherent subsheaf on each of the subsheaves Fi ֒→ F , and with given isomorphisms of SL(2, C)-equivariant sheaves Fi /Fi−1 ∼ = Ei ⊠ O(2i) ,

0 ≤ i ≤ m,

for some sheaves Ei on X, with trivial SL(2, C)-action. Note again that sheaf filtrations and equivariant sheaf filtrations form an abelian category in an obvious way (cf. Definitions 1.2 below). We are now ready to state our main result in this section. Theorem 1.1. There is a one-to-one correspondence between the set of indecomposable coherent SL(2, C)-equivariant sheaves, up to twisting by powers of q ∗ O(1), on X × P1 , and the indecomposable objects in any of the following equivalent categories: (i) The category of SL(2, C)-equivariant sheaf filtrations on X × P1 . (ii) The category of sheaf chains on X. The subobjects and quotient objects are also in one-to-one correspondence. Proof. Most of the work has already been done above. We only need to explain how to associate a chain on X to an SL(2, C)-equivariant sheaf filtration F as (1.13). To show this, let E(≤i) = Fi |X for 0 ≤ i ≤ m. These form a filtration of P -equivariant sheaves on X, E (≤•) : 0 ֒→ E(≤0) ֒→ E(≤1) ֒→ · · · ֒→ E(≤m) = E , E(≤i) /E(≤i−1) ∼ = Ei ⊗ M2i ,

0 ≤ i ≤ m,

where E := F|X . The H-action on the extension 0 −→ E(≤0) −→ E(≤1) −→ E1 ⊗ M2 −→ 0 0 ↓ ↓ ↓ 2 id 0 −→ E(≤0) −→ E(≤1) −→ E1 ⊗ M2 −→ 0

makes E(≤1) to split as a C∗ -equivariant sheaf into its weight subsheaves (cf. Proposition 1.1) as E(≤1) ∼ = E0 ⊗ M 0 ⊕ E1 ⊗ M 2 , while the X − -action on E(≤1) must be given by a morphism φ1 : E1 → E0 (again because [H, X − ] = −2X − ). For the other SL(2, C)-equivariant subsheaves Ei , for

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i > 1, we may now proceed by induction, so a similar argument shows that the SL(2, C)-equivariant extension 0 −→ E(≤i−1) −→ E(≤i) −→ Ei ⊗ M2i −→ 0 , splits, as a C∗ -extension, into E(≤i) ∼ = E(≤i−1) ⊕ Ei ⊗ M2i ∼ =

M j≤i

Ej ⊗ M2j

while the X − -action on E(≤i) is given by the morphisms φj : Ej → Ej−1 , j < i, already describing the X − -action on E(≤i−1) , plus a new morphism φi : Ei → Ei−1 . For i = m we get the sheaf chain (1.9). The correspondence between the morphisms of these categories is straightforward. 2. Holomorphic Filtrations One of the main aims of this paper is to study natural gauge theory equations and stability criteria for equivariant bundles on X × P1 , to consider later their dimensional reduction to X. As we will show, these are natural generalizations of the familiar Hermitian–Einstein equation and stability for holomorphic vector bundles, which we briefly review in the sequel. 2.1. Hitchin–Kobayashi correspondence for holomorphic vector bundles Let F be a holomorphic vector bundle over a compact n-dimensional K¨ahler manifold (M, ω), whose volume is normalized to 2π. A hermitian metric h on F defines ¯ is compatia unique unitary connection Ah on (F , h) whose associated ∂-operator ble with the holomorphic structure of F ; let Fh be its curvature. We say that h is Hermite–Einstein if it satisfies √ −1ΛFh = µ(F )idF , (2.15) where Λ is contraction with the K¨ ahler form of M , and µ(F ), is the slope of F which, for any torsion free coherent sheaf F ′ , is defined as µ(F ′ ) = Here ′

deg F = ′

′

Z

M

deg F ′ . rk F ′

c1 (F ′ ) ∧

ω n−1 (n − 1)!

is the degree of F , where c1 (F ) is the first Chern class of F ′ , and rk F ′ is the rank of F ′ . The existence of a Hermitian–Einstein metric is related to the algebro-geometric condition of stability by the so-called Hitchin–Kobayashi correspondence. Recall that a holomorphic vector bundle F is said to be (semi)stable if for every proper

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coherent subsheaf F ′ ֒→ F, µ(F ′ ) < (≤)µ(F ), where proper means that 0 < rk F ′ < rk F ). A direct sum of stable holomorphic vector bundles, all of them with the same slope, is called polystable. The Hitchin–Kobayashi correspondence states that the existence of a Hermite–Einstein metric on F is equivalent to the polystability of F . For Riemann surfaces, this is equivalent to the theorem of Narasimhan and Seshadri [32] (reproved using gauge-theoretic methods in [13]), and for higher dimensions is the theorem of Donaldson, Uhlenbeck and Yau [14, 15, 38] (see also [27, 29]). 2.2. Deformed Hermite–Einstein equations and stability for holomorphic filtrations When we consider the dimensional reduction to X of an invariant Hermitian– Einstein metric on an SL(2, C)-equivariant vector bundle over X × P1 we obtain some vortex-type equations over the corresponding holomorphic chain over X. One immediately realizes that these equations should naturally depend on as many real parameters as morphisms in the chain, while from the Hermitian–Einstein equation we only get one parameter, which is encoded in the K¨ ahler form on X × P1 . As mentioned in the introduction, the solution to this problem is given by the fact that any equivariant holomorphic vector bundle F on X × P1 has a natural SL(2, C)equivariant holomorphic filtration F , as in (1.13), with as many steps as morphisms are in the corresponding holomorphic chain. One must consider a certain deformation of the Hermite–Einstein equation on F , which is naturally associated to the holomorphic filtration F . This deformation generalizes previous equations associated to holomorphic extensions [9, 17]. The results of this section have been generalized in [30] by considering the deformation of the Hermite–Einstein equation naturally associated to any K¨ahler fibration associated to principal bundle. However, it is only the deformation associated to the filtration F that one needs for the dimensional reduction of an equivariant vector bundle F on X × P1 . Let F be a holomorphic filtration over a compact K¨ ahler manifold (M, ω) given by (1.14). The deformed Hermite–Einstein equation involves as many parameters as steps are in the filtration, and has the form   τ0 I0   τ1 I1 √   −1ΛFh =  (2.16) , . ..   τm Im

where Fh is the curvature of the Chern connection on F associated to a hermitian metric h on F , and the RHS is a diagonal matrix, with constants τ0 , τ1 , . . . , τm ∈ R, written in blocks corresponding to the splitting which a hermitian metric h defines in the filtration F . Taking traces in (2.16) and integrating over M , we see that the parameters are constrained by

Dimensional Reduction, Equivariant Bundles and Stable Chains m X i=0

171

τi rk(Fi /Fi−1 ) = deg F ,

which means that there are only m independent parameters, as one would expect. If τ0 = · · · = τm = µ(F ), (2.16) reduces of course to the Hermite–Einstein equation (2.15). We shall write (2.16) in the slightly different way given in the following. Definition 2.1. Let τ = (τ0 , . . . , τm ) ∈ Rm+1 , and let F , as in (1.14), be a F :F →F holomorphic filtration on M . Given a hermitian metric h on F , let πh,i be the (smooth) h-orthogonal projection onto the k-orthogonal (smooth) vector F be the h-orthogonal projection subbundle of Fi−1 in Fi , for 1 ≤ i ≤ m, and let πh,0 onto F0 in F . We say that h satisfies the τ -Hermite–Einstein equation on F if m X √ F −1ΛFh = τi πh,i . (2.17) i=0

If such a τ -Hermite–Einstein metric h exists on F , we say that F is a τ -Hermite– Einstein holomorphic filtration.

If we take the m independent parameters among τ0 , . . . , τm to be α0 , . . . , αm−1 ∈ R defined by αi = τi+1 − τi

(0 ≤ i ≤ m − 1) ,

(2.18)

then τi = µα (F ) −

m−1 X j=i

αj

(0 ≤ i ≤ m − 1) ,

τm = µα (F ) ,

(2.19)

and the τ -Hermite–Einstein equation is m−1 X √ −1ΛFA = µα (F )idF − αi πhFi .

(2.20)

i=0

Here πhFi : F → F is the h-orthogonal projection onto Fi , for 0 ≤ i ≤ m, and (according to Definition 2.3 below) Pm−1 αi rk Fi . µα (F ) = µ(F ) + i=0 rk F As in the ordinary Hermite–Einstein equation, the existence of solutions to the deformed equation is related to a stability condition for the holomorphic filtration. Definition 2.2. Let F , as in (1.14), be a holomorphic filtration on M . (a) A sheaf subfiltration of F is a sheaf filtration ′ = F′ , F ′ : 0 ֒→ F0′ ֒→ F1′ ֒→ · · · ֒→ Fm

where F ′ is a subsheaf of F , such that Fi′ = Fi ∩ F ′ for 0 ≤ i ≤ m. (b) The sheaf subfiltration F ′ ֒→ F is called proper if 0 < rk F ′ < rk F .

(2.21)

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(c) The sheaf filtration F is called decomposable if it can be written as a direct sum F = F (1) ⊕ F (2) of sheaf subfiltrations with F (1) 6= F , F (2) 6= F , the direct sum being defined in the obvious way. Otherwise, F is called indecomposable. (d) The sheaf filtration F is called simple if its only endomorphisms (i.e. the endomorphisms f of F with f (Fi ) ⊂ Fi for 0 ≤ i ≤ m) are the multiples of the identity endomorphism. Definition 2.3. Let α = (α1 , . . . , αm ) ∈ Rm , and let F , as in (1.14), be a sheaf filtration on M . We define its α-degree and α-slope respectively by degα F = deg F +

m−1 X

αi rk(Fi ) ,

i=0

µα (F ) =

degα F . rk F

We say that the sheaf filtration F is α-(semi)stable if for all proper sheaf subfiltrations F ′ ֒→ F , we have µα (F ′ ) < (≤)µα (F ). A α-polystable sheaf filtration is a direct sum of α-stable sheaf filtrations, all of them with the same α-slope. Semistable and stable filtrations have the usual properties of other already studied (semi)stable objects (existence of destabilizing filtrations, Harder–Narasimhan and Jordan–H¨ older filtrations, stable ⇒ simple, etc.) (cf. e.g. [28]). In the next section we will prove a Hitchin–Kobayashi correspondence establishing the precise relation between the deformed Hermite–Einstein equation and the stability condition for a filtration. 2.3. Hitchin–Kobayashi correspondence for filtrations Theorem 2.1. Let α = (α0 , . . . , αm−1 ) ∈ Rm be an m-tuple of positive real numbers, and define τ = (τ0 , . . . , τm ) ∈ Rm+1 by Eq. (2.19). A holomorphic filtration F is τ -Hermite–Einstein if and only if it is α-polystable. Proof. Let F be a holomorphic filtration, given by (1.14), and let ∂¯F be the Dolbeault operator on F defining its holomorphic structure. Let h be a τ -Hermite– Einstein metric on F , i.e. h satisfies Eq. (2.20). We can assume that F is indecomposable. To prove that it is α-stable, let F ′ ֒→ F , given as in (2.21), be a proper sheaf subfiltration. As usual with stability conditions, we can assume that F ′ is saturated, i.e. F /F ′ is torsion-free, hence a reflexive sheaf filtration (cf. e.g. [28, ′ V.5.22]). Let πhF be the weak holomorphic vector subbundle of (F , h) associated to the reflexive subsheaf F ′ (cf. e.g. [6, Sec. 3.11]). The weak holomorphic vector subbundle of (F , h) associated to the reflexive subsheaf Fi′ ֒→ F (0 ≤ i ≤ m) is ′ F′ then πh i := πhF ◦ πhFi . Therefore  Z √ ′ 1 1 1 ′ F′ kβ(πhF )k2L2 ,h ( tr(π −1ΛFh )) − µα (F ) = h ′ rk F 2π M 2π +

m−1 X i=0

αi rk Fi′



,

(2.22)

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′ ′ ′ ′ where β(πhF ) = πhF ∂¯F (πhF ) = ∂¯F (πhF ) is the second fundamental form of F ′ in (F , h) (cf. e.g. [6, Secs. 2.5 and 3.11 ]). From (2.20), we have almost everywhere on M (cf. e.g. [6, Sec. 3.11]) ! m−1 m−1 X X ′ ′ √ ′ F αi πh i = µα (F )rk F ′ − αi rk(Fi′ ) . tr(πhF ( −1ΛFh )) = tr µα (F )πhF −

i=0

i=0

Therefore ′

2 F 1 kβ(πh )kL2 ,h < µα (F ) , µα (F ) = µα (F ) − 2π rk F ′ ′

′

since the indecomposibility of F implies β(πhF ) 6= 0. To prove the converse, we make use of methods which have become standard in this kind of correspondences [6, 15, 33, 34, 38]; we shall basically follow [6, 33, 34]. We can assume that F is α-stable, which in particular implies that it is simple. It is convenient to decompose the τ -Hermite–Einstein Eq. (2.20), into its trace and trace-free parts. One obtains the equivalent equations 1√ −1Λ tr Fh = µ(F ) , r ! m−1 m−1 X X √ 1 ◦ αi ri idF − αi πkFi , −1ΛFh = r i=0 i=0

(2.23)

where r = rk F , ri = rk Fi , and Fh◦ is the trace-free part of the curvature Fh . Let k be a smooth hermitian metric on F whose determinant det(k) is a hermi√ Vr F verifying −1ΛFdet(k) = λ(F )r (such a hermitian tian metric on det(F ) = metric k exists by Hodge theory). This equation is the trace-part of the τ -Hermite– Einstein equation in the form (2.23). Let χ = det(k), and let S ◦ (k) = {s ∈ Ω0 (End F )|s∗k = s, tr s = 0} and Met(χ) = {h = kes |s ∈ S ◦ (k)} , which is the space of metrics with determinant χ. Fix an integer p > 2n, and define the Lp2 -Sobolev completions Lp2 S ◦ (k), Metp2 (χ) of these spaces to be Metp2 (χ) = {kes |s ∈ Lp2 S ◦ (k)} . We shall now define a modified Donaldson lagrangian Mα , suitable for the trace-free Eq. (2.23). Let MD (k, h) be the Donaldson lagrangian, defined by Donaldson [14] on pairs of metrics k, h on F (see e.g. [6, Sec. 3.2] for the notation). The principal properties of MD are as follows (see e.g. [14] and [6]). Given hermitian metrics h, j, k on F , MD (k, j) + MD (j, h) = MD (k, h) .

(2.24)

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Along any path ht = kets ∈ Metp2 , s ∈ Lp2 S(k), we have: Z √ d MD (k, ht ) = 2 −1 tr(sΛFht ) , dt M d (ΛFt ) = Λ∂¯F ∂h (s) = △′h (s) , dt t=0 Z d2 MD (k, ht ) = 2 |∂¯F s|2ht . dt2 M

(2.25) (2.26) (2.27)

Here ∂¯h = ∂¯Ah , where Ah is the Chern connection of (∂¯F , h), and △′h = ∂¯F ∂¯h∗ + ∂¯h∗ ∂¯F . We shall also need the invariant R1 (k, h) = log det(k −1 h) = tr log(k −1 h) associated to a pair of hermitian metrics k, h on any vector bundle (cf. [14, Sec. 1.2]). This satisfies R1 (k, j) + R1 (j, h) = R1 (k, h)

(2.28)

for any metrics h, j, k. The modified Donaldson lagrangian is Mα (k, h) = MD (k, h) + 2

Z

m−1 X

αi R1 (ki , hi ) .

M i=0

Here k, h are hermitian metrics on F and ki , hi are the induced hermitian metrics on the vector subbundles Fi , for 0 ≤ i ≤ m, by k, h, respectively. From (2.24) and (2.28) it follows that Mα (h, j) + Mα (j, k) = Mα (h, k) .

(2.29)

Define √ mα (h) = ΛFh◦ + −1

"

m−1 1 X αi ri r i=0

!

idF −

m−1 X

αi πhFi

i=0

#

for any hermitian metric h on F . Note that mα (h) is trace-free and that mα (h) = 0 is precisely the trace-free part of Eq. (2.23). Proposition 2.1. If s ∈ Lp2 S ◦ (k), and ht = kgt ∈ Metp2 (χ), gt = ets , then Z √ d Mα (k, ht ) = 2 −1 tr(s ◦ mα (ht )); (a) dt M (b)

(c)

d2 Mα (k, ht ) ≥ 0 ; dt2 d2 = 0 if and only if s = 0 . M (k, h ) α t dt2 t=0

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Proof. Given v, w ∈ Fi in the same fibre,

(v, w)ht,i = (v, w)ht = (ets v, w)k = (πkFi ets v, w)k = (πkFi ets v, w)ki

so ki−1 ht,i = πkFi ets |Fi .

(2.30)

Let ut,i = πhFti : Fi⊥ → Fi be the restriction of πhFti s to the ht -orthogonal complement Fi⊥ of Fi in F . Then R1 (ki , ht,i ) = tr log(πkFi ets |Fi ) = t tr(πkFi s) +

t2 tr(ut,i u∗t,i ) + o(t3 ) , 2

so d R1 (ki , ht,i ) = tr(πhFt s) , dt d2 R1 (ki , ht,i ) = |ut,i |2ht . dt2 From (2.25) and (2.31), it follows that ! Z X √ m−1 √ d αi πhFi tr s(ΛFh − −1 Mα (k, ht ) = 2 −1 dt M i=0 Z √ tr(s ◦ mα (ht )) , = 2 −1

(2.31) (2.32)

M

since s is trace-free. This proves (a). From (2.27) and (2.32), we also get m−1 X d2 2 ¯ αi kut,ik2L2 ,ht . M (k, h ) = 2k ∂ sk + 2 2 α t F L ,ht dt2 i=0

(2.33)

This proves (b). To prove (c), let s be such that d2 Mα (k, ht )/dt2 = 0 at t = 0. From (2.33), it follows that ∂¯F s = 0 and ui = 0 for 0 ≤ i ≤ m − 1. Therefore s is a holomorphic endomorphism of the filtration F . Since F is simple and s is trace-free, it follows that s = 0. We shall use the notation Mα (−) = Mα (k, −). Let B > 0 be such that kmα (k)k2L2 ,k ≤ B. We are interested in the minima of Mα in the set Metp2,B (χ) := {h ∈ Metp2 (χ)| kmα (h)k2Lp ,h ≤ B} .

d Lemma 2.1. Let Lα,h (s) = dt mα (ht )|t=0 , for s ∈ Lp2 S ◦ (h) and h ∈ Metp2 (χ), √ where ht = hets . If F is simple, then −1Lα,h is a bijective h-self-adjoint elliptic operator. √ Pm−1 d αi dt |t=0 πhFti . (cf. e.g. [6, Proof. We know that −1Lα,h (s) = ∆′h (s) + i=0 Eq. (3.19)]). This is obviously an elliptic operator; it is h-self-adjoint since ∆′h is, and ∗h  d Fi ∗ht d Fi d Fi ∗h d Fi π = (πht ) = πht , = (πht ) dt ht t=0 dt dt dt t=0 t=0 t=0

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the second equality is due to the fact that, for ht = hets , πh∗ht t = e−ts (πht )∗h ets = √ πht . Hence by the Hodge theorem, in order to prove that −1Lα,h is bijective it is enough to see that it has no kernel. Now if Lα,h (s) = 0, then Z √ √ d tr(s −1mα (ht )) 0 = (s, −1Lα,h (s))L2 ,h = dt M t=0 √ d2 = (2 −1)−1 2 Mα (k, ht ) , dt t=0 and hence Proposition 2.1(c) gives s = 0.

Proposition 2.2. If F is simple and h ∈ Metp2,B (χ) minimizes Mα in Metp2,B (χ), then mα (h) = 0. Proof. Let s ∈ Lp2 S ◦ (h) and set ht = hets ∈ Metp2 (χ). In principle there are values of t as small as necessary with ht 6∈ Metp2,B (χ) even although h ∈ Metp2,B (χ). Since √ √ −1Lα,h is bijective (Lemma 2.1), there exists s ∈ Lp2 S ◦ (h) with −1Lα,h (s) = √ − −1mα (h) and, as ([6, Eq. (3.4.2)]), for the path ht = hets ∈ Metp2 (χ), we get d2 d M (h, h )| = −2 α t t=0 dt dt2 Mα (h, ht )|t=0 . Let us assume that s 6= 0. We will show that this leads to a contradiction. From Proposition 2.1(b–c), it follows that d Mα (h, ht ) < 0. (2.34) dt t=0 We know that h minimizes Mα in Metp2,B (χ), and hence (2.34) implies that for every ǫ > 0 exists t ∈ (−ǫ, ǫ) with ht ∈ Metp2,B (χ), i.e. kmα (ht )kpLp ,ht > B. But √ √ √ −1Lα,h is bijective =⇒ −1mα (h) = − −1Lα,h (s) 6= 0 ,

d kmα (ht )kpLp ,ht |t=0 = −kmα (h)kpLp ,h < 0, so and as ([6, Eq. (3.4.2)]), we obtain dt that exists ǫ > 0 such that for every t ∈ (−ǫ, ǫ) with t 6= 0, we have kmα (ht )kpLp ,ht < kmα (h)kpLp ,h ≤ B, hence ht ∈ Metp2,B (χ) for t ∈ (−ǫ, ǫ), leading to a contradiction. We thus have that s = 0, which implies mα (h) = −Lαh (s) = 0.

Proposition 2.3. There exist positive constants C1 , C2 such that for all h = kes ∈ Metp2,B (χ), sup |s|k ≤ C1 + C2 Mα (kes ) , or otherwise F is not α-stable. Proof. The proof closely parallels [33, 34] and [6], so we shall only give some indications. Firstly, there is an equivalence between C 0 and L1 bounds for s ∈ Lp2 S ◦ (k) such that h = kes ∈ Metp2 (χ), as in ([6, Sec. 3.7]). Secondly, we shall assume that it is not possible to find constants C1 , C2 verifying the inequalities in Proposition 2.3. Then, as in ([6, Sec. 3.8]), there is a sequence of positive constants {Ca }∞ a=0 with ∞ Ca → ∞, and a sequence {sa }a=1 with ksa kL2 ,k → ∞ and ksa k ≥ Ci Mα (kesa ). Set

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la = ksa kL1 ,k and ua = sa /la . Then kua kL1 ,k = 1, so the equivalence of the C 0 and L1 bounds gives sup |ua | ≤ C(B). Proposition 2.4. By taking a subsequence, ua −→ u∞ weakly in L21 S ◦ (k). The : R × R −→ R be endomorphism u∞ is nontrivial and satisfies the following: let 1 any smooth positive function which satisfies (x, y) ≤ x−1 whenever x > y. Then Z Z √ h (u∞ )∂¯F (u∞ ), ∂¯F (u∞ )ik + tr(u∞ −1mα (k)) ≤ 0 .

F

F

M

F

M

Proof. We know that ksa kL1 ≥ Ca Mα (kesa ) with la = ksa kL1 → ∞. The expression for Mα (kes ), s ∈ Lp2 S(k) is (cf. (2.30) for the last term) Z m−1 Z Z X √ s σi tr(sπkFi ) , hΨ(s)∂¯F s, ∂¯F sik + 2 tr(s −1ΛFk ) + 2 Mα (ke ) = 2 M

M i=0

M

with Ψ : R × R −→ R defined by

ey−x − (y − x) − 1 , Ψ(x, y) = (y − x)2

(see Lemma 5.2.1 in [34]). Hence the inequality ksa kL1 ≥ Ca Mα (kesa ) can be written Z 1 1 −1 la ua C ≥ hΨ(la ua )∂¯F (ua ), ∂¯F (ua )ik Mα (ke ) = la 2 a 2li M ! Z Z m−1 X √ tr ua αa πkFa . + tr(ua −1ΛFk ) + M

M

i=0

The rest is as in ([34, Proposition 6.3.3]).

In order to prove Proposition 2.3, we now see that F is not α-stable. As in ([6, Lemma 3.9.2]), the eigenvalues of u∞ are constant almost everywhere; we list them in ascending order as λ0 < λ1 < · · · < λl , and let pj : R → R be smooth functions such that, for 0 ≤ j ≤ l − 1, pj (x) = 1 if x ≤ λj , pj (x) = 0 if x ≥ λj+1 , and for j = l, pl (x) = 1 if x ≤ λl . Define πj′ = pj (u∞ ), which, as in ([6, Sec. 3.11]), are weak holomorphic vector subbundles of (F , k), hence they represent reflexive subsheaves Fj′ of F , which form a sheaf filtration F ′ : 0 ֒→ F0′ ֒→ F1′ ֒→ · · · ֒→ Fl′ = F .

Let ′ ′ ′ ֒→ Fj1 ֒→ · · · ֒→ Fjm , F ′j : 0 ֒→ Fj0

for 0 ≤ j ≤ l, be the sheaf subfiltration of F induced by Fj′ (cf. Definition 2.6.1(c)). ′ are then reflexive, for Fj′ is reflexive. In this way we Its component subsheaves Fji have obtained a sequence of sheaf subfiltrations 0 ֒→ F ′0 ֒→ F ′1 ֒→ · · · ֒→ F ′l = F F′

F′

′ ֒→ F. Let πk ji = πk j ◦ πkFi = of F , all of them formed by reflexive subsheaves Fji

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´ L. Alvarez-C´ onsul & O. Garc´ia-Prada F′

′ πkFi ◦ πk j be the weak holomorphic subbundle of (F , k) associated to Fji . As in ([6, Eq. (3.7.2)]),

u∞ =

l X

′ λj πjF

=

j=0

l X

λj (πj′

j=0

−

′ πj−1 )

= a idF −

l−1 X

aj πj′

(2.35)

j=0

with aj = λj+1 − λj > 0 for 0 ≤ j ≤ m − 1, and a = λl . Taking traces (cf. [6, Eq. (3.6.a)]) a rk F =

l−1 X j=0

aj rk Fj′ ,

(2.36)

since tr u∞ = 0. Multiplying by πkFi , u∞ πkFi = aπkFi − traces again, tr(u∞ πkFi )

= a rk Fi −

l−1 X i=0

Pl−1

j=0

F′

aj πk ji , and taking

′ aj rk Fji .

(2.37)

Using the result and notation of [34] or ([6, Proposition 2.3]), + * l−1 l−1 X X aj ∂¯F (π ′ ), ∂¯F (π ′ ) aj |∂¯F (π ′ )|2 = j

j

k

j

j=0

j=0

=

* l−1 X

k

+

aj (dpj )2 (u∞ )∂¯F (u∞ ), ∂¯F (u∞ )

j=0

F (u

=h

F

¯

¯

∞ )∂F (u∞ ), ∂F (u∞ )ik

k

.

(2.38)

F

Pl−1 : R × R −→ R is defined by = j=0 aj (dpj )2 , hence it satisfies the Here conditions of Proposition 2.4. We make use of this proposition and (2.35)–(2.38) to estimate the number Q(α) := a degα (F ) −

l−1 X

aj degα (F ′j ) ,

j=0

and to obtain a contradiction if F is α-stable. On one hand, if F is α-stable, then µα (F ′j ) < µα (F ) for 0 ≤ j ≤ l − 1, and using (2.36), we see that l−1 1 X aj (rk(Fj′ ) degα (F ) − rk(E) degα (F ′j )) Q(α) = rk F j=0

=

l−1 X j=0


aj rk(Fj′ ) µα (F ) − µα (F ′j ) > 0 .

(2.39)

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On the other hand, the expression in ([6, Eq. (3.68b)]) for the degree of the reflexive subsheaf Fj′ ֒→ F associated to the weak holomorphic vector subbundles πj′ , Z Z √ ′ ′ tr(πj −1ΛFk ) − 2π deg Fj = |∂¯F (πj′ )|2k , (2.40) M

M

together with Eqs. (2.35), (2.37) and (2.38), give Z Z ¯ ¯ h (u∞ )∂F (u∞ ), ∂F (u∞ )ik + 2πQ(α) = M

F

M

√ tr(u∞ −1mα (k)) ,

since tr u∞ = 0. Therefore Proposition 2.4 implies 2πQ(α) ≤ 0, in contradiction with the inequality (2.39). This proves the main estimate (Proposition 2.3). The rest of the proof follows exactly as ([6, Sec. 3.14]) in or ([34, Sec. 6.6]). For one-step holomorphic filtrations (m = 1), this theorem has been proved in [9], and [17]. 2.4. Equivariant holomorphic filtrations Coming back to the SL(2, C)-manifold M = X × P1 we will consider here an equivariant version of the Hitchin–Kobayashi correspondence for filtrations studied above. Let F be an SL(2, C)-equivariant holomorphic filtration on X × P1 as in (1.13). By taking SL(2, C)-invariant sheaf subfiltrations of F and SU(2)-invariant hermitian metrics on F we can consider the corresponding invariant versions of Definitions 2.1–2.3. The following theorem gives the precise relation of SL(2, C)invariant α-stability of SL(2, C)-equivariant holomorphic filtrations and α-stability of ordinary Holomorphic filtrations. Theorem 2.2. Let F be a SL(2, C)-equivariant holomorphic filtration on X × P1 . F is SL(2,C)-invariantly α-stable if and only if it is SL(2,C)-invariantly indecomposable and, considered as a holomorphic filtration, has a direct sum decomposition into α-stable holomorphic filtrations F a , 0 ≤ a ≤ N, which are the transformed of one of them, say F 0 , by elements ga ∈ SL(2, C): F=

N M a=0

Fa ,

F a = F g0a ,

ga ∈ SL(2, C) ,

1≤a≤N.

(2.41)

Here F ga denotes the transformed of F 0 by ga ∈ SL(2, C). Proof. The proof closely parallels that of ([18, Theorem 6]), where a similar result is given for SL(2, C)-invariant stable holomorphic vector bundles. We shall hence only indicate the changes that are necessary to adapt that proof to the present case. We first show, by using a destabilising saturated sheaf subfiltration on F , that if F is SL(2, C)-invariantly α-stable, then it is α-semistable. Then we show

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that, if it is α-semistable but not α-stable, it has a proper saturated α-stable sheaf subfiltration F ′ ′ F ′ : 0 ֒→ F0′ ֒→ F1′ ֒→ · · · ֒→ Fm = F′ .

This means that 0 < rk F ′ < rk F , µα (F ′ ) < µα (F ) and F ′ is α-stable. Finally it is proved that F ′ is a holomorphic subfiltration of F and that F decomposes as a direct sum of F ′ and other holomorphic subfiltrations transformed of F ′ by different elements of SL(2, C). By combining the above theorem with Theorem 2.1, one gets an equivariant version of the Hitchin–Kobayashi correspondence, whose proof is similar to that of ([18, Theorems 4 and 5]), and will be thus omitted. Theorem 2.3. Let F be a SL(2, C)-equivariant holomorphic filtration on X × P1 . Let α = (α0 , . . . , αm−1 ) ∈ Rm be an m-tuple of positive real numbers, and let τ = (τ0 , . . . , τm ) ∈ Rm+1 be given by Eq. (2.19). Then F is SL(2, C)-invariantly α-polystable if and only if it is SU(2)-invariantly τ -Hermite–Einstein. 3. Holomorphic Chains In this section X is a compact K¨ ahler manifold of complex dimension n with volume normalized to vol(X) = 2π. 3.1. Vortex equations and stability for holomorphic chains Definition 3.1. Let C = (E, φ) be a holomorphic chain over X as given by (1.9). Let τ = (τ0 , τ1 , . . . , τm ) ∈ Rm+1 . Let h = (h0 , h1 , . . . , hm ) be an (m + 1)-tuple of hermitian metrics, where hi is a metric on Ei . We say that h satisfies the chain τ -vortex equations if √ 1 −1ΛFh0 + φ1 ◦ φ∗1 = τ0 idE0 , 2 √ 1 (3.42) −1ΛFhi − (φ∗i ◦ φi − φi+1 ◦ φ∗i+1 ) = τi idEi , (1 ≤ i ≤ m − 1) , 2 √ 1 −1ΛFhm − φ∗m ◦ φm = τm idEm . 2 These equations admit an interpretation, as a zero moment map condition, for the action of the group G0 × · · · × Gm on an appropriate configuration space generalizing that of triples [19], where Gi is the unitary gauge group of (Ei , hi ). The proof is similar to that of ([19, Lemma 2.2]). Definition 3.2. Let C = (E, φ) a holomorphic chain, as in (1.9). (a) A sheaf subchain of C is a sheaf chain ′

C :

′ φm Em −→

φ′m−1 ′ Em−1 −→

φ′

1 · · · −→ E0′ ,

(3.43)

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such that Ei′ is a subsheaf of Ei for 0 ≤ i ≤ m, and φi ◦ fi = fi−1 ◦ φ′i for 1 ≤ i ≤ m, where fi : Ei′ ֒→ Ei are the inclusion morphisms. Pm Pm (b) The sheaf subchain C ′ ֒→ C is called proper if 0 < i=0 rk Ei′ < i=0 rk Ei . (c) The holomorphic chain C is called decomposable if it can be written as a direct sum C = C (1) ⊕ C (2) of holomorphic subchains with C (1) 6= C, C (2) 6= C. Otherwise, C is called indecomposable. (d) The holomorphic chain C is called simple if its only endomorphisms are the multiples λ idC of the identity endomorphism. Definition 3.3. Let C be a holomorphic chain as in (1.9). Let α = (α0 , α1 , . . . , αm ) ∈ Rm . The α-degree and α-slope of a coherent sheaf chain C ′ , are defined by ′

degα C =

m X i=0

deg Ei′

−

m X i=0

αi rk Ei′ ,

deg C ′ µα (C ′ ) = Pm α ′ , i=0 rk Ei

respectively. We say that the coherent sheaf chain C is α-(semi)stable if for all proper sheaf subchains C ′ ֒→ C, µα (C ′ ) < (≤)µα (C). A direct sum of α-stable coherent sheaf chains, all of them with the same α-slope, is called α-polystable. Remarks. (a) If we translate the parameter vector α by a global constant c ∈ R, obtaining α′ = (α′0 , . . . , α′m ), with α′i = αi + c, then µα′ (C) = µα (C) − c. Hence the stability condition does not change under global translations. This means that there are only m independent parameters among α0 , . . . , αm . (b) If C has an (m + 1)-tuple of metrics satisfying the chain τ -vortex equations then, taking traces in (3.42), integrating over X, and summing for 0 ≤ i ≤ m, Pm one sees that the τ -parameters are constrained by the relation i=0 deg Ei = Pm i=0 τi rk Ei . This equation can also be written as degτ (C) = 0 .

(3.44)

As in the case of α-stability, this means that there are only m independent parameters among τ0 , . . . , τm . In the proof of the Hitchin–Kobayashi correspondence, we shall take τ satisfying (3.44) for the equations but, for the stability condition, it will be convenient to use α = (α0 , . . . , αm ), defined by αi = τi − τ0 ,

(0 ≤ i ≤ m) ,

(3.45)

so that α0 = 0 and the independent parameters are α1 , . . . , αm . Recall that τ (semi)stability is then equivalent to α-(semi)stability. As a result, µα (C) = µτ (C)+ τ0 = τ0 , so that τ is given, in terms of α1 , . . . , αm , by τ0 = µα (C) , τi = αi + µα (C) ,

(1 ≤ i ≤ m) .

(3.46)

As sheaf filtrations, semistable and stable chains have the usual properties of other already studied (semi)stable objects (existence of destabilizing filtrations, Harder–Narasimhan and Jordan–H¨ older filtrations, stable ⇒ simple, etc.).

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3.2. Hitchin–Kobayashi correspondence Theorem 3.1. Let C be a holomorphic chain. Let τ = (τ0 , τ1 , . . . , τm ) ∈ Rm+1 be such that degτ (C) = 0. The holomorphic chain C admits an (m + 1)-tuple h = (h0 , h1 , . . . , hm ) of hermitian metrics satisfying the chain τ -vortex equation if and only if it is τ -polystable. Proof. First of all, let α = (α0 , . . . , αm ) be defined by (3.45). Recall that τ -(poly)stability is equivalent to α-(poly)stability (cf. Remarks in Sec. 3.1). Suppose that h satisfies the chain τ -vortex, Eq. (3.42). We can assume that C is indecomposable. We have to prove that it is α-stable. Let C ′ ֒→ C, given as in (3.43), be a proper saturated coherent sheaf subchain (i.e. each Ei /Ei′ is torsion-free), which must consist of reflexive subsheaves Ei′ ֒→ Ei . Let πi′ be the weak holomorphic vector subbundle of (Ei , hi ) associated to the reflexive subsheaf Ei′ . In a similar way to Theorem 2.1, one has 1 µα (C ′ ) = Pm ′ i=0 rkEi !  X Z m  m X √ 1 1 ′ 2 ′ ′ kβ(πi )kL2 ,hi − tr(πi ( −1ΛFhi )) − × αi rk Ei 2π X 2π i=0 i=1

¯ of Ei . From (3.42), where β(πi′ ) = πi′ ∂¯Ei (πi′ ) = ∂¯Ei (πi′ ), and ∂¯Ei is the ∂-operator one obtains √ 1 tr(πi′ ( −1ΛFhi )) = τi rk Ei′ + tr(πi′ ◦ (φ∗i ◦ φi − φi+1 ◦ φ∗i+1 )) , 2 almost everywhere in X, for 0 ≤ i ≤ m. Let ′ ◦ φi ◦ πi′ , φ′i = πi−1

′′ φ′′i = πi−1 ◦ φi ◦ πi′′ ,

′ ′′ φ⊥ i = πi−1 ◦ φi ◦ πi

′′ (with πi′′ = id − πi′ ). Since C ′ is a subchain of C, πi−1 ◦ φi ◦ πi′ = 0, so ′′ ′′ ′′ φi = φ′i ◦ πi′ + φ⊥ i ◦ πi + φi ◦ πi ,

′ ⊥∗ ′ ′′∗ ′′ φ∗i = φ′∗ i ◦ πi−1 + φi ◦ πi + φi ◦ πi .

Then a straightforward computation shows that Z X m m X 2 ′ ∗ ∗ kφ⊥∗ tr(πi ◦ (φi ◦ φi − φi+1 ◦ φi+1 )) = − i kL2 ,hi ∗hi−1 . X i=0

i=0

(where k · kL2 ,hi ∗hi−1 is the L2 -norm on the space of smooth sections Ω0 (Hom(Ei , Ei−1 )) of the vector bundle Hom(Ei , Ei1 ) of smooth homomorphisms from Ei to Ei−1 with respect to the metric hi ∗ hi−1 induced by hi and hi−1 on Hom(Ei , Ei−1 )). Therefore, since Vol(X) = 2π, µα (C ′ ) =

2π ×

1 Pm

i=0

rk Ei′

m m X X 1 ⊥ 2 ′ ′ 2 (2πτi rk Ei − kφi kL2 ,hi ∗hi−1 − kβ(πi )kL2 ,hi ) − 2π αi rk Ei′ 2 i=0 i=1

!

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which from (3.46), gives  m  X 1 1 ⊥ 2 1 ′ 2 P µα (C ) = µα (C) − kβ(πi )kL2 ,hi + kφi kL2 ,hi ∗hi−1 . ′ 2π m 2 i=0 rk Ei i=0 ′

Hence µα (C ′ ) < µα (C) since the indecomposibility of C implies β(πi′ ) 6= 0 or φ⊥ i 6= 0 for some i. To prove the converse we can assume that C is α-stable, which in particular implies that it is simple. We first write the chain τ -vortex Eq. (3.42) in a different manner, suitable for a decomposition into a trace and a trace-free part. Let (3.47) E = E 0 ⊕ · · · ⊕ Em , Pm Pm with rank and degree r := rk E = i=0 ri , deg E = i=0 deg Ei . Let πi : E → Ei be Pm ¯ of E. Finally, let the canonical projections, and ∂¯E = i=0 ∂¯Ei ◦πi be the ∂-operator φ=

m X i=0

φi ◦ πi ∈ Ω0 (End E)

(3.48)

be the global endomorphism of E defined by the Higgs fields φi . Any (m + 1)-tuple h = (h0 , . . . , hm ) of hermitian metrics, where hi is a metric on Ei , gives rise to a hermitian metric h = h 0 ⊕ · · · ⊕ hm ,

(3.49)

on E whose associated connection Ah has curvature Fh = (3.42) is equivalent to m X √ 1 −1ΛFh − (φ∗ ◦ φ − φ ◦ φ∗ ) = τi πi , 2 i=0

Pm

i=0

Fhi ◦ πi , and hence

Ei , Ej h-orthogonal for i 6= j .

(3.50)

√ P The trace part of the LHS of (3.50) is simply −1Λ tr Fh , while the RHS is m i=0 τi ri = µ(E) (cf. (3.44)). We denote by Fh◦ the trace-free part of the curvature Fh ; the term φ∗ ◦ φ − φ ◦ φ∗ is trace-free itself, while the trace-free part of the RHS is (cf. (3.46)) m m X X µ(E) λ(E) idE = idE µα (C)πi − αi πi + τi πi − r r i=0 i=1 i=0

m X

m X

1 αi πi − = r i=1 Therefore (3.50) is equivalent to 1√ −1Λ tr Fh = µ(E) r

m X

αi ri

i=1

m X √ 1 ∗ 1 ◦ ∗ −1ΛFh − (φ ◦ φ − φ ◦ φ ) = αi πi − 2 r i=1

!

idE .

m X i=1

αi ri

!

idE ,

Ei , Ej h-orthogonal for i 6= j .

(3.51)

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Let k = k0 ⊕ · · · ⊕ km be a hermitian metric on E = E0 ⊕ · · · ⊕ Em , whose deV terminant hermitian metric χ = det(h) on det(E) = r E verifies the first equation √ in (3.51), i.e. −1ΛFχ = λ(E)r. Such a hermitian metric always exists, since it can be obtained by a conformal transformation k = k ′ ea , a ∈ C ∞ (X), from any ′ , by solving an equation for a; this equation has a solution other k ′ = k0′ ⊕ · · · ⊕ km because of the Hodge theorem. We write ( ) m X si ◦ πi si ∈ Ω0 (End Ei ) (0 ≤ i ≤ m), s∗k = s, tr(s) = 0 , S ◦ (k) = s = i=0

so the set of hermitian metrics h on E such that det(h) = χ and Ei , Ej are horthogonal for i ≤ j, is Met(χ) = {h = kes |s ∈ S ◦ (k)} .

Fix an integer p > 2n, and define the Lp2 -Sobolev completions Lp2 S ◦ (k), Metp2 (χ) of these spaces, so that Metp2 (χ) = {h = kes |s ∈ Lp2 S ◦ (k)}. Observe that each s ∈ Lp2 S ◦ (k) gives rise to a hermitian metric h in Metp2 (χ), by s =

m X i=0

si ◦ πi ,

h = kes = h0 ⊕ · · · ⊕ hm ,

with si ∈ Ω0 (End Ei ) ,

(3.52)

with hi = ki esi .

We shall consider the modified Donaldson lagrangian Mφ,α , defined as Z X m 2 2 τi tr(log(ki−1 hi )) , Mφ,α (k, h) = MD (k, h) + (kφkL2 ,h − kφkL2 ,k ) − 2 X i=0

where k, h ∈ Metp2 (χ), and ki , hi are the hermitian metrics on Ei given by the decompositions k = k0 ⊕ · · · ⊕ km , h = h0 ⊕ · · · ⊕ hm . It is obvious that Mφ,α (k, h) + Mφ,α (h, j) = Mφ,α (k, j)

(3.53)

for any metrics k, h, j. (cf. e.g. [6, the proof of Lemma 3.3.2(ii)]). Define ! ! √ m m X X √ −1 1 (φ∗h ◦ φ − φ ◦ φ∗h ) + −1 αi ri idE αi πi − mφ,α (h) = ΛFh◦ + 2 r i=1 i=1

where φ∗h ∈ Ω0 (End E) denotes the adjoint of φ with respect to the hermitian metric h. (So mφ,α is trace-free, and the second equation in (3.51) is mφ,α (h) = 0.) Proposition 3.1. If s ∈ Lp2 S ◦ (k), and ht = kgt ∈ Metp2 (χ), gt = ets , then Z √ d Mφ,α (k, ht ) = 2 −1 tr(s ◦ mφ,α (ht )) ; (a) dt X (b) (c)

d2 Mφ,α (k, ht ) ≥ 0 ; dt2 d2 = 0 if and only if s = 0 . M (k, h ) φ,α t dt2 t=0

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Before we prove this proposition, note that, as in Eq. (3.52), the endomorphisms ts ∈ Lp2 S ◦ (k), for t ∈ R, give rise to hermitian metrics ht in Metp2 (χ), by m X si ◦ πi , with si ∈ Ω0 (End Ei ) , s = (3.54) i=0 ht = kets = h0,t ⊕ · · · ⊕ hm,t ,

with hi,t = ki etsi .

Proof. (a) We know that φ∗ht = e−ts φ∗k ets and ki−1 hi,t = tsi , so that Z X Z m ∗ht ∗k τi tr(si ) . tr(φ ◦ φ − φ ◦ φ ) − 2t Mφ,α (k, ht ) = MD (k, ht ) + X

X i=0

It is now easily seen that (cf. e.g. [6, Eq. (3.16)]) ! Z m X √ d Mφ,α (k, ht ) = τi tr(si ) . tr(2s −1ΛFht ) − s(φ∗ht φ − φφ∗ht ) − 2 dt X i=0 √ R Since tr s = 0, this is precisely 2 −1 X tr(smφ,α (ht )). (b) Recall that d ΛFht = Λ∂¯E ∂ht (s) dt where dAht = ∂ht + ∂¯E is the connection associated to the hermitian metric ht on E (cf. e.g. [6, Eq. (3.19)]). Since φ∗ht = e−ts φ∗h ets , we have √ d −1 mφ,α (ht ) = Λ∂¯E ∂ht (s) + (−sφ∗ht φ + φ∗ht sφ + φsφ∗ht − φφ∗ht s) , (3.55) dt 2 and hence Z  d2 Mφ,α (k, ht ) = 2 tr(sΛ∂¯E ∂ht (s)) dt2 X  1 ∗ht ∗ht ∗ht ∗ht (3.56) + tr(ssφ φ − sφ sφ − sφsφ + sφφ s) . 2 √ Since −1[Λ, ∂¯E ] = ∂h∗ht t (cf. [6, Eq. (1.18a)]), and s∗ht = s, we get Z √ −1 tr(sΛ∂¯E ∂ht (s)) = k∂ht (s)k2L2 ,ht . X

If we set u = φ ◦ s and v = s ◦ φ in (3.56), so that u∗ht = sφ∗ht , v ∗ht = φ∗ht s, we have   1 d2 2 2 2 Mφ,α (k, ht ) = 2 k∂ht (s)kL2 ,ht + (kukL2 ,ht + kvkL2 ,ht − 2(u, v)L2 ,ht ) . dt2 2 (3.57) R This is a real number, for (u, v)L2 ,ht = (v, u)L2 ,ht = X tr(vu∗ht ) where, since R s∗ht = s, we have tr(vu∗ht ) = tr(uv ∗ht ), and therefore (u, v)L2 ,ht = X tr(uv ∗ht ) = (u, v)L2 ,ht ∈ R. Moreover, we see that 2(u, v)L2 ,ht ≤ 2|(u, v)|L2 ,ht ≤ 2kukL2,ht kvkL2 ,ht ≤ kuk2L2 ,ht + kvk2L2 ,ht ,

and hence (3.57) gives the result.

(3.58)

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d (c) From (3.57) and (3.58), we see that dt 2 Mφ,α (k, ht ) = 0 at t = 0 if and only if ∂h s = 0 and 2(u, v)L2 ,h = 2kukL2,h kvkL2 ,h = kuk2L2 ,h + kvk2L2 ,h . The first condition is equivalent to ∂¯E (s) = 0, since s = s∗h ⇒ ∂h (s)T = ∂¯E (s). The second condition

happens if and only if u = v, i.e. φ ◦ s = s ◦ φ, or equivalently φi ◦ si = si−1 ◦ φi for 1 ≤ i ≤ m, if and only if s = (s0 , . . . , sm ) : C → C is an endomorphism of chains. If C is simple, this must be s = c idC for some constant c, i.e. si = c idEi (0 ≤ i ≤ m), Pm and the condition tr( i=0 si ◦ πi ) = 0 gives c = 0, i.e. s = 0. Corollary 3.1. If h = kes with s ∈ Lp2 S ◦ (k), then hφ∗h ◦ φ − φ ◦ φ∗h , sih ≤ hφ∗k ◦ φ − φ ◦ φ∗k , sik . Proof. We define f (t) = −hφ∗ht ◦ φ − φ ◦ φ∗ht , siht , for ht = kets , t ∈ R. In the d f (t) = |u|2ht + |v|2ht − 2(u, v)ht ≥ 0 with proof of Proposition 3.1, we showed that dt u := φ ◦ s, v := s ◦ φ. Therefore f (1) ≥ f (0). We shall use the notation Mφ,α (−) = Mφ,α (k, −). Choose a real number B > 0 such that kmφ,α (k)k2L2 ,k ≤ B. We are interested in the minima of Mα in the set Metp2,B (χ) := {h ∈ Metp2 (χ)| kmφ,α (h)k2Lp ,h ≤ B} . Lemma 3.1. If C is simple, then the map Lφ,α,h : Lp2 S ◦ (h) → Lp2 S ◦ (h) defined d mφ,α (ht ), where ht = hets , s ∈ Lp2 S ◦ (h), is for any h ∈ Metp2 (χ) by Lφ,α,h (s) = dt √ such that −1Lφ,α,h is a bijective h-self-adjoint elliptic operator.

√ Proof. From (3.55), we see that −1Lφ,α,h (s) = ∆′h (s) + Uφ,h (s), with Uφ,h : Lp2 S ◦ (k) → Lp2 S ◦ (k) given by Uφ,h = 21 (sφ∗h φ − φ∗h sφ − φsφ∗h + φφ∗h s). We √ can easily see that this is a h-self-adjoint operator, and hence −1Lφ,α,h is an elliptic h-self-adjoint operator. By the Hodge theorem, in order to prove that it is bijective it is enough to see that it has no kernel. Now, if Lφ,α,h (s) = 0, we R √ d2 d have 0 = (s, Lφ,α,h (s))L2 ,h = dt X tr(s −1mφ,α (ht ))|t=0 = dt2 Mφ,α (ht )|t=0 , and hence Proposition 3.1(c) implies s = 0. Proposition 3.2. If C is simple and h ∈ Metp2,B (χ) minimizes Mφ,α in Metp2,B (χ), then mφ,α (h) = 0. Proof. This is analogous to Proposition 2.2. Proposition 3.3. There exist positive constants C1 , C2 such that for all h = kes in Metp2,B (χ), sup |s|k ≤ C1 + C2 Mφ,α (kes ) or otherwise C is not α-stable. Proof. The proof is similar to that of Proposition 2.3, so we will only give some indications. Firstly, there is an equivalence between C 0 and L1 bounds for s ∈

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Lp2 S ◦ (k) such that h = kes ∈ Metp2 (χ) ; its proof needs a result analogous to ([6, Proposition 3.7.1]), for mφ,α , and this is proved similarly to [6] once Corollary 3.1 is used. Secondly, we assume that it is not possible to find constants C1 , C2 verifying the inequalities in Proposition 3.3. Hence there is a sequence of positive constants {Ca }∞ a=0 with Ca → ∞, and a sa 2 with ks k → ∞, ks k ≥ C M (ke ) and φ ◦ ua = ua ◦ φ. sequence {sa }∞ a L ,k a i φ,α a=1 ∗k ∗k Observe that if φ ◦ s = s ◦ φ, then φ ◦ s = s ◦ φ , so φ∗h = e−s φ∗k es = φ∗k for h = kes , hence kφkL2 ,h = kφkL2 ,k ; this cancels two terms in the expression of Mφ,α (h). Set la = ksa kL1 ,k and ua = sa /la , so kua kL1 ,k = 1, and the equivalence of the C 0 and L1 bounds gives sup |ua | ≤ C(B). Proposition 3.4. After taking a subsequence, ua −→ u∞ weakly in L21 S ◦ (k). The : R × R −→ R be endomorphism u∞ is nontrivial and satisfies the following: let 1 any smooth positive function which satisfies (x, y) ≤ x−1 whenever x > y. Then ! Z Z Z m X √ αi πi ≤ 0 , h (u∞ )∂¯E (u∞ ), ∂¯E (u∞ )ik − tr u∞ tr(u∞ −1ΛFk ) +

F

F

X

X

0

F

where u∞,i ∈ Ω (End Ei ) for 0 ≤ i ≤ m define u∞ = bundle endomorphism u∞ verifies φ ◦ u∞ = u∞ ◦ φ.

X

Pm

i=0

i=1

u∞,i ◦ πi . The limiting

Proof. We know that ksa kL1 ≥ Ca Mφ,α (kesa ) with la = ksa kL1 → ∞. Hence the inequality ksa kL1 ≥ Ca Mφ,α (kesa ) can be written (cf. the proof of Proposition 2.1) Z 1 −1 1 la ua C ≥ hΨ(la ua )∂¯E (ua ), ∂¯E (ua )ik Mφ,α (ke ) = la 2 a 2la X Z X Z m √ 1 τi tr(ua,i ) , + tr(ua −1ΛFk ) − 2 X i=0 X P with ua,i ∈ Ω0 (End Ei ) for 0 ≤ i ≤ m defining ua = m i=0 ua,i ◦ πi . The rest is as in ([6, Proposition 3.9.1]). Obviously φ ◦ u∞ = u∞ ◦ φ, for φ ◦ sa = sa ◦ φ. In order to prove Proposition 3.3, we now see that C is not α-stable. The eigenvalues of u∞ are constant almost everywhere; we list them in ascending order as λ0 < λ1 < · · · < λl , and let pj : R → R be smooth functions such that, for 0 ≤ j ≤ l − 1, pj (x) = 1 if x ≤ λj , pj (x) = 0 if x ≥ λj+1 , and for j = l, pl (x) = 1 if x ≤ λl . Define πj′ = pj (u∞ ), which, as in ([6, Sec. 3.11]), are weak holomorphic vector subbundles of (E, k), hence they represent nested reflexive subsheaves Fj′ of F , which form a sheaf filtration F ′ : 0 ֒→ F0′ ֒→ F1′ ֒→ · · · ֒→ Fl′ = F .

Now let, for 0 ≤ j ≤ m,

′ ′ ֒→ · · · ֒→ Eli′ = Ei , ֒→ E1i F ′i : 0 ֒→ E0i

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be the sheaf subfiltration of F ′ induced by the holomorphic vector subbundle Ei of ′ ′ = Ej′ ∩ Ei for 0 ≤ j ≤ l, 0 ≤ i ≤ m. Each Eji is obviously a reflexive sheaf. E, i.e. Eji Lm Lm The direct sum decomposition E = i=0 Ei gives another one F ′ = i=0 F ′i for the filtrations, so that ′ ′ ′ ⊕ Ej1 ⊕ · · · ⊕ Ejm Ej′ = Ej0

for 0 ≤ j ≤ l .

(3.59)

′ = πj′ ◦ πi = πi ◦ πj′ be the weak holomorphic vector subbundle of (E, k) Let πji Pm ′ ′ , so that (3.59) implies πj′ = i=0 πji . associated with the reflexive subsheaf Eji Pl ′ As in (2.35), one has u∞ = j ′ =0 λj ′ πjF′ . From this and u∞ ◦ φ = φ ◦ u∞ it is straightforward to get πj′ ◦ φ ◦ πj′ = φ ◦ πj′ . This means that the image of Ej′ by φ is again in Ej′ , i.e. that φ′j := φ|Ej′ : Ej′ → Ej′ is an endomorphism of Ej′ for 0 ≤ j ≤ l−1. ′ ′ , so ) = φ(Ei ∩ Ej′ ) ⊂ Ei−1 ∩ Ej′ = Ej,i−1 But φ(Ei ) ⊂ Ei−1 , for 1 ≤ i ≤ m, hence φ′j (Eji ′ ′ ′ ′ ′ ′ that φj defines, by restriction, homomorphisms φji = φj |Eji = φi |Eji : Eji → Ej,i−1 . Summarizing, we have obtained a sequence of nested reflexive sheaf chains

0 ֒→ C ′0 ֒→ C ′1 ֒→ · · · ֒→ C ′l = C , of C, explicitly given by φ′jm

φ′j,m−1

φ′j1

′ ′ ′ −→ Ej,m−1 −→ · · · −→ Ej0 . C ′j : Ejm

The rest of the proof follows exactly as in Sec. 2.1, where the analogue of (2.36)– Pl−1 Pl−1 ′ , and (2.38) are a rk E = j=0 aj rk Ej′ , tr(u∞ πi ) = a rk Ei − i=0 aj rk Eji l−1 X j=0

F (u

aj |∂¯E (πj′ )|2k = h

¯

¯

∞ )∂E (u∞ ), ∂E (u∞ )ik

,

which are used to estimate the number Q(α) := a degα (C) −

l−1 X

aj degα (C ′j ) .

j=0

Proposition 3.4 gives a contradiction with α-stability. This proves the main estimate (Proposition 3.3). The rest of the proof follows exactly as ([6, Sec. 3.14]) or ([34, Sec. 6.6]). The above theorem gives a direct proof of the Hitchin–Kobayashi correspondence for triples (m = 1) given in [10]. 3.3. Holomorphic chains and variations of Hodge structure In this section we disgress a bit to show the important role played by the moduli space Mchains of semistable holomorphic chains in the study of the topological properties of the moduli space Homred (π1 (X), G)/G of reductive representations of the fundamental group of X on a Lie group G. We shall see that for G = SL(r, C), α-polystable holomorphic chains correspond, for particular values of α, to those

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representations of the fundamental group of X which come from complex variations of Hodge structure (cf. [35]). As a consequence, a more detailed study of Mchains is of great interest. Let G = SL(r, C) and let X be a closed compact Riemann surface of genus g ≥ 2. By non-abelian Hodge theory there is an equivalence between Homred (π1 (X), G)/G and the moduli space M of polystable Higgs bundles on X. From a holomorphic point of view, these are pairs consisting of a holomorphic vector bundle V and a holomorphic morphism Φ : V → V ⊗ K, where K is the canonical bundle on X. The appropriate (poly)stability condition involves the slope µ(V) = d/r (d is the degree and r is the rank of V). Recall that (V, Φ) is (semi)stable if any holomorphic vector subbundle V ′ ⊂ V with Φ(V ′ ) ⊂ V ′ ⊗ K has slope µ(V ′ ) < (≤)µ(V). Hitchin [24] (see also [33] for higher dimensions) proves that M can be seen as the moduli space of pairs (A, Φ) associated to a smooth complex vector bundle V with hermitian metric h, where A is a unitary connection on V and Φ ∈ Ω1,0 (End0 V ), satisfying Hitchin’s self-duality equation Fh◦ + [Φ, Φ∗ ] = 0

(3.60)

(End0 means trace free endomorphisms, and Fh◦ is the trace free part of the curvature). We shall say that the holomorphic Higgs bundle has a harmonic metric if there is a hermitian metric h on V whose associated connection Ah satisfies this self-duality equation. When (d, r) = 1, Hitchin proves that M is a smooth hyperk¨ ahler manifold. He also computes the Poincar´e polynomial of M for the rank 2 case, and similar results for rank 3 bundles have been obtained by Gothen [22]. This moduli space has a circle action which respects the symplectic form of one of the complex structures. The map f : M → R, defined by f (A, Φ) = kΦk2 , is a perfect Morse function. Therefore, to compute the Poincar´e polynomial of M, it is crucial to describe the critical submanifolds of f . In Proposition 3.5 below they are described as moduli spaces of α-stable holomorphic chains, for particular values of α. This is similar to ([24, Proposition 7.1]) and ([22, Propositions 2.6, 3.10]). One expects that an analysis similar to [37] will provide the Poincar´e polynomials of the moduli space of these α-stable holomorphic chains. This requires considering the whole range of values of α, and should be useful for the computation of the Poincar´e polynomials of M for Higgs bundles of arbitrary rank, as the rank 3 case points out (in this case one finds as critical manifolds moduli spaces of stable triples E1 → E0 and chains of line bundles L2 → L1 → L0 , cf. [22]). To state Proposition 3.5, we start with an argument in ([25, Sec. 7]) (see also [22]): f is a moment map for the circle action, so the critical points are the fixed points. A pair (A, Φ) represents a fixed point if and only if there is an infinitesimal gauge transformation ψ ∈ Ω0 (X, End0 (V, h)) such that dA ψ = 0 and [ψ, Φ] = √ −1Φ. Thus the holomorphic structure V defined by the connection A on V splits √ into V = ⊕λ∈∆ V λ , where ψ acts by −1λ on V λ , and ∆ is a sequence of real numbers λ0 < · · · < λm with λi −λi−1 = 1. With respect to this decomposition, Φ = ⊕λ∈∆ Φλ , with Φλ : V λ → V λ+1 ⊗ K, and all these maps are nontrivial. Conversely

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any stable Higgs bundle with this kind of splitting represents a critical point of f (cf. [25]). To express the critical Higgs bundle (V, Φ) in terms of a holomorophic chain, it is convenient to define Vi = V λm−i and Φi = Φλm−i : Vi → Vi−1 ⊗ K, for m 0 ≤ i ≤ m, so that V = ⊕m i=0 Vi and Φ = ⊕i=1 Φi . Let Ei = Vi ⊗ K m−i for 0 ≤ i ≤ m. Then Ei−1 = (Vi−1 ⊗ K) ⊗ K m−i , so it makes sense to define the morphism φi = Φi ⊗ idK m−i : Ei = Vi ⊗ K m−i → Ei−1 = (Vi−1 ⊗ K) ⊗ K m−i , and we get a holomorphic chain C = (E, φ). Proposition 3.5. Let α = (α0 , . . . , αm ) be defined by αi := (m − i) deg K = (m−i)(2g −2) for 0 ≤ i ≤ m. The critical Higgs bundle (V, Φ) is stable (respectively semistable) if and only if the holomorphic chain C is α-stable (respectively αsemistable). Proof. First of all, note that deg(Vi ) = deg(Ei ) − αi rk(Ei ), so µ(V) = µα (C). Assume that (V, Φ) is (semi)stable. We have to show that C is α-(semi)stable. Given ′ i−m ⊂ V is Φ-invariant. a subchain C ′ = (E ′ , φ), the subbundle V ′ = ⊕m i=0 Ei ⊗ K P m ′ ′ ′ i−m ) = degα (C ) and µα (C ′ ) = µ(V′ ) < (≤) Hence deg(V ) = i=0 deg(Ei ⊗ K µ(V) = µα (C). The converse is more delicate, because the Φ-invariant subbundles of V do not necessarily split as V does. We shall give a proof similar to ([23, Lemma 2.2]). Assume that C = (E, φ) is α-(semi)stable. We have to show that (V, Φ) is (semi)stable. Let πi : V → Vi be projection onto the ith factor. To simplify the notation, the projection V ⊗ K → Vi ⊗ K will also be denoted by πi . Let V ′ ⊂ V be a Φ-invariant subbundle. We have to show that µ(V ′ ) < (≤)µ(V). To do this, we shall define a filtration ′ ′ ′ ⊂ V(≤1) ⊂ · · · ⊂ V(≤m) = V′ 0 ⊂ V(≤0)

(3.61)

of coherent subsheaves ′ ⊂ V(≤i)

M j≤i

Vj ,

(3.62)

such that there are short exact sequences ′ ′ −→ V(≤i) −→ Vi′ −→ 0 , 0 −→ V(≤i−1)

(3.63)

′ ′ := V(≤m) with Vi′ a subsheaf of Vi , for 0 ≤ i ≤ m + 1. Here we have defined V(≤m+1) ′ ′ ′ ′ and V(≤−1) = 0 (so Vm+1 = 0 and V0′ = V(≤0) by (3.63)). Start defining V(≤m) = V ′. ′ ′ ⊂ ⊕j≤i Vj and Vi+1 ⊂ Vi+1 have been For i ≤ m, assume that the sheaves V(≤i) ′ defined so that they satisfy the properties above. Let V(≤i−1) and Vi′ respectively be ′ ′ : V(≤i) → Vi , so there is a short exact sequence the kernel and the image of πi |V(≤i)

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′ ′ (3.63). Note that, since V(≤i) ⊂ ⊕j≤i Vj , we get V(≤i−1) ⊂ Ker(πi ) ∩ ⊕j≤i Vj = ′ ⊕j≤i−1 Vj , while Vi ⊂ Im(πi ) = Vi . ′ ⊗ K for 0 ≤ i ≤ m. Because of the short Next we prove that Φ(Vi′ ) ⊂ Vi−1 ′ ′ ) ⊂ V(≤i−1) ⊗ K. So let v ′ exact sequences (3.63), it is enough to see that Φ(V(≤i) ′ ′ be a germ of a section in the stalk V(≤i),x of the sheaf V(≤i) at some point x ∈ X. Then v ′ ∈ ⊕j≤i Vj,x so Φ(v ′ ) ∈ ⊕j≤i−1 Vj,x ⊗ Kx since Φ takes each factor Vj into Vj−1 ⊗ K; therefore πi (Φ(v ′ )) = 0, so ′ ⊗ Kx Φ(v ′ ) ∈ V(≤i−1),x

′ ′ ′ by the definition of V(≤i−1) . This proves that Φ(V(≤i) ) ⊂ V(≤i−1) ⊗ K. Defining ′ ′ ′ m−i ′ , we see that φi (Ei ) ⊂ Ei−1 , hence these subsheaves define a sheaf E i = Vi ⊗ K ′ subchain C of C. By α-(semi)stability of C, µα (C ′ ) < (≤)µα (C). From (3.63), we get m m m X X X ′ ′ ′ ′ deg Vi′ = degα (C ′ ) , rk Ei , deg V = rk Vi = rk V = i=0

i=0

i=0

′

since deg Vi′ = deg Ei′ − αi rk Ei′ , so µ(V ′ ) = µα (C ) < (≤)µα (C) = µ(V).

Notice that one can also show directly a correspondence between (3.60) and (3.42). 4. Dimensional Reduction We have proved Hitchin–Kobayashi correspondences for the two different categories describing SL(2, C)-equivariant holomorphic vector bundles on X × P1 . In this section we will prove that these two correspondences are actually equivalent. More precisely, we will show that one can be obtained from the other by dimensional reduction. To show how the chain vortex equations are dimensional reduction of the modified Hermitian–Einstein equation on the corresponding filtration, we first study invariant connections on X × P1 . We give a gauge-theoretic interpretation of the Higgs fields and reprove, in terms of Dolbeault operators, the equivalence between SL(2, C)-equivariant holomorphic vector bundles (up to a twist), and holomorphic chains on X (Theorem 1.1). We close the circle by showing the equivalence between the stability conditions for a holomorphic chain and the corresponding filtration. In this section X is a compact K¨ahler manifold of complex dimension n. We normalize the volumes to Vol(P1 ) = 1 and Vol(X) = 2π. 4.1. Dimensional reduction of invariant connections The group SU(2) acts on X × P1 by the trivial action on X and the standard action on P1 via the identification P1 = SU(2)/U (1). Our goal now is to study SU(2)-invariant connections on a given SU(2)-equivariant smooth hermitian vector bundle on X × P1 . To do this, we first recall some basic facts (cf. [19]) concerning the structure of such a bundle.

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We shall denote E ′ ⊠ E ′′ = p∗ E ′ ⊗ q ∗ E ′′ , for any pair of bundles E ′ on X and E ′′ on P1 , where p : X × P1 → X, q : X × P1 → P1 are the canonical projections. Proposition 4.1. Let F be an SU(2)-equivariant vector bundle on X × P1 , and h be an SU(2)-invariant hermitian metric on F. Then (a) Every SU(2)-equivariant smooth complex vector bundle F on X × P1 can be equivariantly decomposed, uniquely up to isomorphism, as m m M M F˜i = Ei ⊠ H ⊗ni (4.64) F = i=0

i=1

with F˜i = Ei ⊠ H ⊗ni , where Ei is a smooth complex vector bundle on X, with trivial SU(2)-action, H is the line bundle on P1 with Chern class 1, and ni ∈ Z are all different. (b) The vector bundles F˜i are SU(2)-invariantly orthogonal to each other; in other Lm ˜ ˜ ˜ ˜ words, h = i=0 h i , with hi an SU(2)-invariant metric on Fi . Moreover, hi = p∗ hi ⊗ q ∗ h′i , where hi is a metric on Ei , and h′i is an SU(2)-invariant metric on H ⊗ni .

Proposition 4.2. Let (F, h) be an SU(2)-equivariant hermitian vector bundle as given by the proposition above. Let A be a unitary connection on (F, h). Define the operators βji : Ω0 (F˜i ) → Ω1 (F˜j ), corresponding to the covariant derivative dA : Ω0 (F ) → Ω1 (F ) with respect to the decompositions m m M M 0 0 ˜ 1 Ω (Fi ) , Ω (F ) = Ω1 (F˜i ) . Ω (F ) = i=0

i=0

˜ i ), and if i 6= j, βji ∈ Then βii = dA˜i for some unitary connection A˜i on (F˜i , h Ω1 (Hom(F˜i , F˜j )) is the adjoint of −βij , that is, h(βji s, t) + h(s, βij t) = 0

Pm P for s ∈ Ω0 (F˜i ), t ∈ Ω0 (F˜j ). We can therefore write dA = i=0 dA˜i ◦ πi + j
Let (F, h) be an SU(2)-equivariant hermitian vector bundle as in Proposition 4.1, and let A and G be the space of unitary connections and the unitary gauge group of (F, h), respectively. The group SU(2) acts on both of them by the rule dγ(A) = γ ◦ dA ◦ γ −1 ,

γ(g) = γ ◦ g ◦ γ −1 ,

for A ∈ A, g ∈ G, γ ∈ SU(2). Let ASU(2) ⊂ A the subset of SU(2)-invariant connections, and the subgroup G SU(2) ⊂ G of SU(2)-invariant gauge transformations. These two sets can be described in terms of objects defined on X. To see this, let (Ei , hi ) for 0 ≤ i ≤ m be the hermitian vector bundles on X appearing in F . Let Ai and Gi be the corresponding spaces of unitary connections and unitary gauge groups. The group Gi acts on both Ωji := Ω0 (Hom(Ei , Ej )) and Ωij = Ω0 (Hom(Ej , Ei )), by gi (φji ) = φji ◦ gi−1 ,

gi (φij ) = gi ◦ φij ,

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for φji ∈ Ωji , φij ∈ Ωij , gi ∈ Gi . There is an induced action of the group G0 ×· · ·×Gm on Y Ωji , N := A0 × · · · Am × (i,j)∈Q

where Q := {(i, j)|0 ≤ i, j ≤ m, ni − nj = 2} . Proposition 4.3. (a) There is a one-to-one correspondence between ASU(2) and N which associates to any element of N given by Ai ∈ Ai for 0 ≤ i ≤ m, and φji ∈ Ωji , for (i, j) ∈ Q, the SU(2)-invariant unitary connection A ∈ ASU(2) given by dA =

m X i=0

dA˜i ◦ πi +

X

(i,j)∈Q

βji ◦ πi −

X

(i,j)∈Q

∗ ◦ πj . βji

(4.65)

˜ i ) given by d ˜ = dp∗ A ⊗ id + id ⊗ dq∗ A′ , Here A˜i is the unitary connection of (F˜i , h i Ai ni where A′ni the unique SU(2)-invariant unitary connection of (H ⊗ni , h′i ), and βji := p∗ φji ⊗ q ∗ α ∈ Ω0,1 (Hom(F˜i , F˜j )) for (i, j) ∈ Q , where α ∈ Ω1P1 (H ⊗−2 ) is an SU(2)-invariant holomorphic (0, 1)-form, which is unique up to a constant factor that we will fix. (b) There is a one-to-one correspondence between G SU(2) and G0 × · · · × Gm . The correspondence associates to any (g0 , . . . , gm ) ∈ G0 × · · · × Gm , the SU(2)Pm invariant unitary gauge transformation of (F, h) g = i=0 g˜i ◦ πi , with g˜i = p∗ gi ∈ ˜ i )) ∼ Ω0 (Aut(F˜i , h = Ω0 (Aut(p∗ Ei , p∗ hi )). (c) These correspondences are compatible with the actions of the groups of (b) 1−1 on the sets of (a), hence there is a one-to-one correspondence ASU(2) /G SU(2) ←→ N /G0 × · · · × Gm . Proof. (a) Fix a unitary connection A◦i on each hermitian vector bundle (Ei , hi ), ˜ i ) by d ˜◦ = dp∗ A◦ ⊗id+id⊗ 0 ≤ i ≤ m. Define unitary connections A˜◦i on each (F˜i , h Ai i dq∗ A′i . These are obviously SU(2)-invariant, so they give rise to an SU(2)-invariant Pm unitary connection A◦ on (F, h), defined by dA◦ = ˜◦ ◦ πi , and all the i=0 dA i SU(2)-invariant unitary connections A on (F, h) take the form A = A◦ + θ for θ ∈ Ω1 (End(F, h))SU(2) , the SU(2)-invariant subset of Ω1 (End(F, h)). Here End(F, h) is the vector bundle of anti-hermitian endomorphisms of (F, h). Let End(Ei , hi ) be the vector bundle of anti-hermitian endomorphisms of (Ei , hi ). Then all we have to prove is that there is a one-to-one correspondence between Ω1 (End(F, h))SU(2) and the set Y Ωji , Θ := Ω1 (End(E0 , h0 )) × · · · × Ω1 (End(Em , hm )) × (i,j)∈Q

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and that this correspondence associates to any element of Θ given by θi ∈ Ω1 (End (Ei , hi )), for 0 ≤ i ≤ m, and φji ∈ Ωji , for (i, j) ∈ Q, the SU(2)-invariant θ ∈ Ω1 (End (F, h))SU(2) given by m X X X ∗ βji ◦ πi − βji ◦ πj , θ˜i ◦ πi + θ= i=0

(i,j)∈Q

(i,j)∈Q

where θ˜i = p∗ θi ⊗ id, and βji = p∗ φji ⊗ q ∗ α for (i, j) ∈ Q. It is clear that Ω (End(F, h)) ∼ = 1

Therefore 1

SU(2)

Ω (End(F, h))

∼ =

m M i=0

m M i=0

˜ i )) ⊕ Ω1 (End(F˜i , h

1

∗

∗

∗

∗ 1

P , with TC∗ P1 SU(2) ∼ 0

Ω1 (Hom(F˜i , F˜j )) .

j
˜ i ))SU(2) ⊕ Ω1 (End(F˜i , h

Now, T (X × P ) ∼ =p T X ⊕q T ˜ i )) Ω1 (End(F˜i , h ∗

M

M

Ω1 (Hom(F˜i , F˜j ))SU(2) .

j
∼ = H ⊗2 ⊕ H ⊗−2 , and hence

= Ω (End(Ei , hi ))

and, when ni 6= nj , we have either

Ω1 (Hom(F˜i , F˜j ))SU(2) = 0

if ni − nj 6= ±2, or

Ω1 (Hom(F˜i , F˜j ))SU(2) ∼ = Ω0 (Hom(Ei , Ej )) ,

if ni − nj = ±2. This proves (a). Lm L (b) We have G ⊂ Ω0 (End(F )) ∼ = i=0 Ω0 (End(F˜i )) ⊕ j6=i Ω1 (Hom(F˜i , F˜j )), L hence G SU(2) ⊂ Ω0 (End(F˜i ))SU(2) ⊕ i6=j Ω0 (Hom(F˜i , F˜j ))SU(2) . By similar arguments to those of (a), we see that Ω0 (Hom(F˜i , F˜j ))SU(2) = 0 for ni 6= nj , and Ω0 (End(F˜i ))SU(2) ∼ = Ω0 (End(Ei )). The result is now obvious. (c) is immediate. We shall now consider integrable connections, i.e. those A with FA0,2 = 0. Let A1,1 be the set of integrable unitary connections on (F, h), and let A1,1 SU(2) be be the set of integrable unitary connections on its SU(2)-invariant subset. Let A1,1 i (Ei , hi ). Proposition 4.4. Let N 1,1 the subset of N consisting of those Ai , φji with Ai ∈ Ai1,1 for 0 ≤ i ≤ m, and ∂¯Aj ∗Ai (φji ) = 0 for (i, j) ∈ Q, where Aj ∗ Ai is the connection on Hom(Ei , Ej ) induced by Ai and Aj . The one-to-one correspondence of Proposition 4.3(a) defines a one-to-one correpondence between N 1,1 and A1,1 SU(2) . Proof. Given A ∈ ASU(2) by (4.65), the integrability condition of A is (∂¯A )2 = 0, where m X X ¯ ∂¯A˜i ◦ πi + βji ◦ πi , ∂A = i=0

(i,j)∈Q

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with ∂¯A˜i = ∂¯p∗ Ai ⊗ id + id ⊗ ∂¯q∗ A′n and βji = p∗ φji ⊗ q ∗ α. Therefore i

(∂¯A )2 =

m X i=0

(∂¯A˜i )2 ◦ πi +

X

(i,j)∈Q

∂¯A˜j ∗A˜i (βji ) ◦ πi +

X

X

(i,j)∈Q (j,k)∈Q

(βkj ∧ βji ) ◦ πi ,

where (∂¯A˜i )2 = p∗ (∂¯Ai )2 ⊗ id ,

∂¯A˜j ∗A˜i (βji ) = p∗ (∂¯Aj ∗Ai (φji )) ⊗ q ∗ α ,

βkj ∧ βji = 0 .

Therefore (∂¯A )2 = 0 if and only if (∂¯Ai )2 = 0 for 0 ≤ i ≤ m, and ∂¯Aj ∗Ai (φji ) = 0 for (i, j) ∈ Q. ¯ The last proposition can equivalently be given in terms of ∂-operators. Let C, Ci , ¯ for 0 ≤ i ≤ m, be the sets of integrable ∂-operators (or holomorphic structures) on E, Ei , respectively, and let G c , Gic be the corresponding complex gauge groups. Consider the SU(2)-action on C and G c given by γ(∂¯F ) = γ ◦ ∂¯F ◦ γ −1 , γ(g) = γ ◦ g ◦ γ −1 , for ∂¯F ∈ C, g ∈ G c and γ ∈ SU(2). The set C SU(2) of SU(2)-invariant holomorphic structures on F is precisely the set of structures of SU(2)-equivariant holomorphic vector bundle on X × P1 with underlying SU(2)-equivariant smooth vector bundle Q c acts on the subset N 0,1 of C0 ×· · ·×Cm × (i,j)∈Q Ωji conF . The group G0c ×· · ·×Gm sisting of ∂¯Ei ∈ Ci , and φji ∈ Ωji , satisfying ∂¯Hom(Ei ,Ej ) (φji ) = 0, where ∂¯Hom(Ei ,Ej ) is the holomorphic structure on Hom(Ei , Ej ) induced by ∂¯Ei and ∂¯Ej . Proposition 4.5. Let F be an SU(2)-equivariant smooth vector bundle on X ×P1 , given by (4.64). (a) There is a one-to-one correspondence between C SU(2) and N 0,1 . This correspondence associates to any element of N 0,1 given by ∂¯Ei ∈ Ci for 0 ≤ i ≤ m, and ¯ ∂¯F ∈ C SU(2) given by φji ∈ Ωji , for (i, j) ∈ Q, the ∂-operator ∂¯F =

m X i=0

∂¯F˜i ◦ πi +

X

(i,j)∈Q

βji ◦ πi ,

(4.66)

with ∂¯F˜i = p∗ ∂¯Ei ⊗id+id⊗q ∗ ∂¯H ⊗ni and βji := p∗ φji ⊗q ∗ α ∈ Ω0,1 (Hom(F˜i , F˜j )) for (i, j) ∈ Q. c . The (b) There is a one-to-one correspondence between G c SU(2) and G0c × · · · × Gm c c correspondence associates to any (g0 , . . . , gm ) ∈ G0 × · · · × Gm , the SU(2)Pm invariant complex gauge transformation of F, g = i=0 g˜i ◦ πi , with g˜i = p∗ gi ∈ Ω0 (Aut(F˜i )) ∼ = Ω0 (Aut(p∗ Ei )). (c) These correspondences are compatible with the actions of the groups of (b) on 1−1 the sets of (a), hence there is a one-to-one correspondence C SU(2) /G c SU(2) ←→ c . N 0,1 /G0c × · · · × Gm

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Proof. (a) Let h be an SU(2)-invariant hermitian metric on F , as in Proposition 4.1. The resut follows from Proposition 4.4 and the fact that the metric h (respectively hi for 0 ≤ i ≤ m) gives a bijection between C SU(2) (respectively Ci ) and A1,1 SU(2) (respectively A1,1 i ). (b) The proof is analogous to that of Proposition 4.3(b). (c) is immediate. In Sec. 1, we gave an algebraic proof of the correspondence between coherent SL(2, C)-equivariant sheaves on X × P1 , SL(2, C)-equivariant sheaf filtrations on X×P1 , and sheaf chains on X. We will give now another proof of this correspondence ¯ for the locally free case in terms of ∂-operators. Let F ′ be an SU(2)-equivariant 1 holomorphic vector bundle on X × P (the action of SU(2) can be extended to an action of SL(2, C)). Its underlying SU(2)-equivariant smooth vector bundle F ′ can be equivariantly decomposed as in Proposition 4.1 by ′

F =

m M

F˜i′ ,

′ F˜i′ = Ei ⊠ H ⊗ni ,

i=0

and its holomorphic structure ∂¯F ′ can be decomposed as in Proposition 4.5 by ∂¯F ′ =

m X i=0

∂¯F˜ ′ ◦ πi + i

X

(i,j)∈Q′

βi′ ◦ πi ,

∂¯F˜ ′ = p∗ ∂¯Ei ⊗ id + id ⊗ q ∗ ∂¯H ⊗n′i , i

′ ′ := p∗ φi ⊗ q ∗ α ∈ Ω0,1 (Hom(F˜i′ , F˜i−1 )) , βji

where Q′ := {(i, j)|n′i − n′j = 2} . If F ′ is indecomposable we can write n′0 = m− , . . . , n′i = m− +2i, . . . , n′m = m− +2m for some integer m− , possibly after reordering n′0 , . . . , n′m . Twisting by q ∗ O(−m− ), we obtain another SU(2)-equivariant holomorphic vector bundle F = F ′ ⊗ q ∗ O(−m− ) on X × P1 , whose underlying SU(2)-equivariant smooth vector bundle is F = F ′ ⊗ Lm q ∗ H ⊗−m− = i=0 F˜i with F˜i = Ei ⊠ H ⊗2i , and whose holomorphic structure is ∂¯F =

m X i=0

∂¯F˜i ◦ πi +

m X i=0

βi ◦ πi ,

(4.67)

∂¯F˜i = p∗ ∂¯Ei ⊗ id + id ⊗ q ∗ ∂¯H ⊗2i

βi : = p∗ φi ⊗ q ∗ α ∈ Ω0,1 (Hom(F˜i , F˜i−1 ))

(1 ≤ i ≤ m) .

Here φi := φi−1,i ∈ Ω0 (Hom(Ei , Ei−1 )) satisfy ∂¯Hom(Ei ,Ei−1 ) (φi ) = 0. It is clear that F defines an equivariant filtration as in (1.14). The holomorphic structures ∂¯Ei define holomorphic vector bundles Ei = (Ei , ∂¯Ei ) for 0 ≤ i ≤ m, and the holomorphic morphisms φi ∈ H 0 (Hom(Ei , Ei−1 )) give us a holomorphic chain (1.9),

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which is indecomposable. This gives one direction of the correspondence between equivariant holomorphic vector bundles on X × P1 and holomorphic chains on X. To see the other direction of the correspondence, we just associate to any holomorphic chain C as above, the SU(2)-equivariant holomorphic vector bundle F with Lm underlying SU(2)-equivariant smooth vector bundle F = i=0 F˜i , F˜i = Ei ⊠ H ⊗2i , and holomorphic structure ∂¯F given by (4.67). 4.2. Dimensional reduction and equations Theorem 4.1. Let F be an SL(2, C)-equivariant holomorphic filtration on X × P1 and let C be the corresponding holomorphic chain on X (cf. Theorem 1.1). Let ′ τ = (τ0 , . . . , τm ), τ ′ = (τ0′ , . . . , τm ) ∈ Rm+1 be related by τi′ = τi + 4πi

(0 ≤ i ≤ m) .

Then F admits an SL(2, C)-invariantly τ ′ -Hermite–Einstein metric if and only if C admits an (m+1)-tuple of hermitian metrics satisfying the chain τ -vortex equation. Proof. Let h be an SU(2)-invariant hermitian metric on F . Let A be the unique unitary connection compatible with the holomorphic structure of F . Of course A must be SU(2)-invariant and hence of the form dA =

m X i=0

dA˜i ◦ πi +

m X i=0

βi ◦ πi , (4.68)

dA˜i = p∗ dAi ⊗ id + id ⊗ q ∗ dA′2i

βi : = p∗ φi ⊗ q ∗ α ∈ Ω0,1 (Hom(F˜i , F˜i−1 ))

(1 ≤ i ≤ m) ,

(compare with (4.67)). Since A is integrable we must have ∂¯Ai−1 ∗Ai (φi ) = 0. The curvature of A is given by X X ∗ (FA˜i − βi∗ ∧ βi − βi+1 ∧ βi+1 ) ◦ πi + dA˜i−1 ∗A˜i (βi ) ◦ πi FA = (dA )2 = i

+

X i

i

(βi−1 ∧ βi ) ◦ πi −

X i

∗ ) ◦ πi + dA˜i+1 ∗A˜i (βi+1

X i

(βi+2 ∧ βi+1 ) ◦ πi . (4.69)

A straightforward computation shows that FA˜i = p∗ FAi ⊗ id + id ⊗ q ∗ FA′2i , βi∗ ∧ βi = p∗ (φ∗i ◦ φi ) ⊗ q ∗ (α∗ ∧ α) ,

∗ βi+1 ∧ βi+1 = p∗ (φi+1 ◦ φ∗i+1 ) ⊗ q ∗ (α ∧ α∗ ) ,

dA˜i−1 ∗A˜i (βi ) = p∗ dAi−1 ∗Ai (φi ) ⊗ q ∗ α , βi−1 ∧ βi = 0 ,

∗ dA˜i+1 ∗A˜i (βi+1 ) = p∗ dAi+1 ∗Ai (φ∗i+1 ) ⊗ q ∗ α , ∗ ∗ = 0. ∧ βi+1 βi+2

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√ The curvature of the connection A′2i of H ⊗2i is FA′2i = −2π −1(2i)ωP1 ∈ Ω1,1 P1 . √ 0,1 One can choose α ∈ ΩP1 (H ⊗−2 ) so that α ∧ α∗ = ( −1/2)ωP1 . We thus have √ m  X −1 ∗ ∗ ∗ p (φi ◦ φi − φi+1 ◦ φ∗i+1 ) p FAi + FA = 2 i=0 ∗

⊗ q ωP 1 −

√ − 4π −1p∗ idEi ⊗ q ∗ ωP1



◦ πi +

m X i=0

(p∗ dAi−1 ∗Ai (φi ) ⊗ q ∗ α) ◦ πi

m−1 X i=1

(p∗ dAi+1 ∗Ai (φi+1 ) ⊗ q ∗ α∗ ) ◦ πi .

(4.70)

Let Λ indistinctly be contraction with the K¨ ahler forms ω, ωP1 and ωM = p∗ ω ⊕ Pm q ∗ ωP1 of X, P1 and M = X ×P1 , respectively. Taking into account that i=0 τi′ πi = Pm ∗ ′ ∗ i=0 (p (τi idEi ) ⊗ q idH ⊗2i ) ◦ πi , we see that √  m m X √ X −1 ∗ ′ ∗ (φi ◦ φi − φi+1 ◦ φ∗i+1 ) τi πi = p ΛFAi + ΛFA + −1 2 i=0 i=0 √ + −1τi idEi



◦ πi ,

and the result follows by comparing this with (2.17) and (3.42). 4.3. Dimensional reduction and stability Theorem 4.2. Let F be an SL(2, C)-equivariant holomorphic filtration on X ×P1 , and let C be the corresponding holomorphic chain on X (cf. Theorem 1.1). Let α = (α0 , . . . , αm ) ∈ Rm+1 , α′ = (α′0 , . . . , α′m−1 ) ∈ Rm be related by α0 = 0 ,

αi =

i−1 X j=0

α′j − 4πi

(1 ≤ i ≤ m) .

Then F is SL(2, C)-invariantly α′ -(semi)stable if and only if C is α-(semi)stable. Proof. First of all, from Theorem 1.1, one knows that the SL(2, C)-invariant sheaf subfiltrations ′ =F, F ′ : 0 ֒→ F0′ ֒→ F1′ ֒→ · · · ֒→ Fm ′ ∼ Fi′ /Fi−1 = Ei′ ⊠ O(2i) ,

1 ≤ i ≤ m,

of F are in one-to-one correspondence with the sheaf subchains ′

C :

′ ′ φm −→ Em

φ′m−1 ′ Em−1 −→

φ′

1 · · · −→ E0′ .

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of C. So let F ′ ֒→ F and C ′ ֒→ C, be proper subobjects in correspondence. Then deg(F ′ ) =

m X i=0

deg(Ei′ ⊠ O(2i)) =

m X

deg(Ei′ ) + 2π

i=0

m X

2i rk(Ei′ ) .

i=1

Therefore degα′ (F ′ ) =

m X i=0

deg(Ei′ ) −

m X i=1



αi rk(Ei′ ) + 

m−1 X j=0



α′j 

m X i=0

!

rk(Ei′ )

,

Pm−1 ′ αj , with a similiar equation for F and C. We and hence µα′ (F ′ ) = µα (C ′ ) + j=0 ′ thus have µα′ (F ) < (≤)µα′ (F ) if and only if µα (C ′ ) < (≤)µα (C). Acknowledgements We would like to thank Alastair King for very useful discussions and for important suggestions [27]. O.G. whishes to thank the Ecole Polytechnique at Palaiseau (France) and the Mathematical Institute of Oxford for their support and warm hospitality. The research of L.A. was supported by the Comunidad Aut´onoma de Madrid (Spain) under a FPI grant. This research has been partly supported by the Spanish MEC through the grant PB95–0185, a FPU grant to O.G. and the Acci´ on Integrada HB1998-0006. Both authors have benefited from workshops organized by the VBAC research group under the European algebraic geometry networks AGE and Europroj, supported by the European Union. References 1. D. N. Akhiezer, Lie group actions in complex analysis, Aspects of Mathematics E27, Vieweg, 1995. ´ 2. L. Alvarez-C´ onsul and O. Garc´ıa-Prada, Dimensional reduction and quiver bundles, preprint. 3. M. F. Atiyah and R. Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A308 (1982), 523–615. 4. M. F. Atiyah and N. J. Hithin, The Geometry and Dynamics of Magnetic Monopoles, Princeton Univ. Press, 1988. 5. S. B. Bradlow, Vortices in holomorphic line bundles over closed K¨ ahler manifolds, Commun. Math. Phys. 135 (1990), 1–17. 6. S. B. Bradlow, Special metrics and stability for holomorphic bundles with global sections, J. Diff. Geom. 33 (1991), 169–214. 7. S. B. Bradlow, G. Daskalopoulos, O. Garc´ıa-Prada and R. Wentworth, Stable augmented bundles over Riemann surfaces, in Vector Bundles in Algebraic Geometry, Durham 1993, eds. N. J. Hitchin, P. E. Newstead and W. M. Oxbury, LMS Lecture Notes Series 208, Cambridge Univ. Press, 1985. 8. S. B. Bradlow, G. Daskalopoulos and R. Wentworth, Birational equivalences of vortex moduli, Topology 35 (1996), 731–748. 9. S. B. Bradlow and O. Garc´ıa-Prada, Higher cohomology triples and holomorphic extensions, Comm. Anal. Geom. 3 (1995), 421–463.

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