HITCHIN–KOBAYASHI CORRESPONDENCE FOR EQUIVARIANT BUNDLES ON X × P1 ´ ´ LUIS ALVAREZ–C ONSUL AND OSCAR GARC´IA–PRADA

Abstract. This paper is concerned with the study of gauge theory, stability and dimensional reduction 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 show that there is a Hitchin–Kobayashi correspondence relating the existence of solutions to a natural deformation of the Hermitian–Einstein equation and an appropriate notion of stability for an equivariant bundle on X × P1 . The invariance of the solutions and the stability condition lead, via dimensional reduction, to a Hitchin–Kobayashi correspondence for vortex type equations and a stability condition for a chain of holomorphic bundles on X linked by homomorphisms. Keywords: Equivariant bundle, stability, Hermitian–Einstein equations, vortex equations, holomorphic chains, Hitchin–Kobayashi correspondence, dimensional reduction.

1. Introduction Let M be a compact K¨ahler manifold and let F be a holomorphic vector bundle over M . It is well-known that a natural differential equation to consider for a Hermitian metric h on F is the Hermitian–Einstein equation, also referred sometimes as the Hermitian–Yang-Mills equation. This says that Fh , the curvature of the Chern connection of h must satisfy √ −1 ΛFh = λI, were Λ is the contraction with the K¨ahler form of M , λ is a real number determined by the topology and I is the identity endomorphism of F. A theorem of Donaldson, Uhlenbeck and Yau [D2, D3, UY], also known as the Hitchin–Kobayashi correspondence establishes that the existence of such metric is equivalent to the stability of F in the sense of Mumford–Takemoto. Suppose now that a compact Lie group K acts on M by isometries so that M/K is a smooth K¨ahler manifold and the action on M can be lift to action on F. One can then apply dimensional reduction techniques to study K-invariant solutions to the Hermitian–Einstein equation on F and the corresponding stability condition to obtain a theory expressed entirely in terms of the orbit space M/K. Many important equations in gauge theory arise in this way [H]. In this paper we give a brief account of this programme for the manifold M = X ×P1 , where X is a compact K¨ahler manifold, and K = SU(2) acts trivially on X and in the standard way on P1 , with the K¨ahler structure of X × P1 defined by the product of that of X and the Fubini-Study metric on P1 . More details, references, and some Date: May 2001. 1

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´ ´ LUIS ALVAREZ–C ONSUL AND OSCAR GARC´IA–PRADA

applications can be found in [AG]. In this paper, however, we take the oportunity to explain in a more thorough way (than is has been done in [AG]) the classification of SL(2, C)-equivariant holomorphic bundles and coherent sheaves over X × P1 —where the action of SU(2) has been extended to its complexification, SL(2, C)). We will indeed need to understand SL(2, C)-equivariant coherent sheaves in order to study Hitchin– Kobayashi correspondences. It turns out that every SL(2, C)-equivariant holomorphic vector bundle on X × P1 admits an equivariant holomorphic filtration by equivariant subbundles, which, in turn, is in one-to-one correspondence with a chain consisting of holomorphic vector bundles Ei on X and morphisms φi : Ei → Ei−1 . The filtered structure of the equivariant bundle on X × P1 dictates (by symplectic arguments) that the right equation to consider is a certain deformation of the Hermitian–Einstein equation which involves as many paremeters as terms are in the filtration (notice that the Hermitian–Einstein equation involves just one parameter, λ, which is actually determined by the topology). We show that there is a Hitchin–Kobayashi correspondence relating this equation to a stability criterion for the filtration, and this gives, via dimensional reduction, a Hitchin–Kobayashi correspondence for the holomorphic chains on X, generalising the theory of triples studied in [G, BG]. Acknowledgements. The second author is a member of the European Networks EDGE and EAGER. Both authors are members of the international research group VBAC. 2. Equivariant bundles, sheaves and holomorphic chains Given a complex Lie group G, a complex G–manifold is a complex manifold M with a holomorphic G–action, i.e. a left G–action by biholomorphisms on M ; a G–equivariant holomophic vector bundle on M is a holomophic vector bundle π : F → M , together with holomorphic G–action σ : G × F → F on its total space which commutes with π, and such that for all (g, p) ∈ G × M , the map σg,p : Ep → Eg·p , from Ep = π −1 (p) into Eg·p = π −1 (g · p), induced by this action, is an isomorphism of vector spaces. Let us now consider M = X × P1 , and G = SL(2, C) acting trivially on X and with action on P1 given by the identification P1 = SL(2, C) /P , where P is the subgroup of lower triangular matrices   ∗ 0 P = ⊂ SL(2, C) . ∗ ∗ An SL(2, C)–equivariant holomophic vector bundle F on X×P1 defines a P –equivariant holomorphic vector bundle i∗ F on X by restriction to the slice i : X ∼ = X × P/P ,→ 1 X × P . Conversely, a P –equivariant holomorphic vector bundle E on X defines by induction an SL(2, C)–equivariant holomophic vector bundle F = SL(2, C) ×P E on X × P1 , which by definition is the quotient of SL(2, C) ×E by the P –action, given by p · (g, e) = (gp−1 , p · e), for p ∈ P , (g, e) ∈ SL(2, C) ×E. The SL(2, C)–equivariant action on F is g 0 · [g, e] = [g 0 g, e], for g 0 ∈ SL(2, C) and [g, e] ∈ F. When X is a single point, this induction process supplies an equivalence between the category of holomorphic homogenous vector bundles on P1 (i.e. SL(2, C)–equivariant vector bundles on P1 ) and the category of holomorphic representations of P . When X is not a point, we can still consider X × P1 as a family of projective lines {x} × P1 ,

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parametrised by the points x ∈ X, and a holomorphic SL(2, C)–equivariant vector bundle F on X × P1 as a family of holomorphic homogeneous vector bundles i∗x F on the projective lines ix : {x} × P1 ,→ X × P1 . Thus, a detailed study of the holomorphic representations of P is important in classifying the SL(2, C)–equivariant holomophic vector bundles on X × P1 . We shall see that P -modules admit two equivalent descriptions, both of them with a counterpart in the case of equivariant bundles on X × P1 . We begin with some notation. The Lie algebra p of P is generated by the matrices     1 0 0 0 − H= ,X = , with [H, X − ] = −2X − . 0 −1 1 0 Let W be a representation of P . Since P is not semisimple, W is not generally semisimple. The action of the complex torus C∗ ⊂ P , is given by a weight decomposition (2.1) W = ⊕λ∈∆(W ) Wλ ⊗ Mλ , Wλ = HomC∗ (Mλ , W ) for λ ∈ Z, ∆(W ) = {λ ∈ Z|Wλ 6= 0}, where Mλ is an irreducible C∗ -module with weight λ (i.e. z·v = z λ v for z ∈ C∗ , v ∈ Mλ ), for λ ∈ Z, Wλ are the multiplicity spaces, and ∆(W ) is the set of weights of W . The rest of the P -module structure of W is determined by the action of X − ∈ p. Since [H, X − ] = −2X − , the action of X − on Wλ ⊗ Mλ is defined by 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

(2.2)

ϕ2

ϕ0

ϕ−2

· · · −→ W2 −→ W0 −→ W−2 −→ · · · ϕ−1 ϕ5 ϕ3 ϕ1 · · · −→ W3 −→ W1 −→ W−1 −→ · · ·

Any P -module is the direct sum of an even part (Wλ = 0 if λ is odd) and an odd part (Wλ = 0 if λ is even), so the set of weights of an indecomposable P -module form a sequence m− , m− + 2, . . . , m+ − 2, m+ of consecutive odd or even integers. Such an indecomposable P -module is given by a sequence Wm− , Wm− +2 , . . . , Wm+ of non-zero finite dimensional vector spaces, and non-zero linear maps ϕλ : Wλ → Wλ−2 between consecutive vector spaces. Twisting by M−m− , we get a representation V = W ⊗M−m− whose C∗ -action is given by (2.3) V = ⊕m i=0 Vi ⊗M2i ,

Vi := Wm− +2i , ∆(V ) = {0, 2, . . . , 2m}, m := (m+ −m− )/2,

and whose X − -action is given by maps φi , which can be represented by the diagram (2.4)

φm

φm−1

φ1

Vm −→ Vm−1 −→ · · · −→ V0 .

This is our first description of a P -module. The second description is given in terms of flags of P -modules. Actually, any indecomposable P -module V which has the structure given by (2.12) and (2.4) (after twisting by some M−m− if necessary), admits a flag (2.5)

V(≤•) : 0 ⊂ V(≤0) ⊂ V(≤1) ⊂ . . . ⊂ V(≤m) , V(≤i) /V(≤i−1) ∼ = Vi ⊗ M2i , 0 ≤ i ≤ m,

where V(≤i) := ⊕j≤i Vj ⊗ M2j are nested P -submodules of V . We need now some definitions to generalize this to equivariant bundles and sheaves on X × P1 .

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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 : Ei → Ei−1 (1 ≤ i ≤ m). This is represented by the diagram (2.6)

φm

φm−1

φ1

C : Em −→ Em−1 −→ · · · −→ E0 .

We say that C is a holomorphic chain if the sheaves Ei are locally free. A morphism C 0 → C between a sheaf chain (2.7)

φ0

φ0m−1

φ0

m 1 0 0 −→ Em−1 −→ · · · −→ E00 , C 0 : Em

and (2.7), is given morphisms fi : Ei0 → Ei for each i, such that φi ◦ fi = fi−1 ◦ φ0i for 1 ≤ i ≤ m. It is immediate that sheaf chains form an abelian category. Let M be a complex G–manifold. By a coherent G–equivariant sheaf on M we will mean a coherent sheaf F together with a holomorphic G–equivariant action on F (cf. e.g. [Ak]). The coherent G–equivariant sheaves, with the G–equivariant morphisms, form an abelian category. A G–equivariant sheaf filtration on M is a finite sequence of G–invariant coherent subsheaves of a coherent G–equivariant sheaf F on M , (2.8)

F : 0 ,→ F0 ,→ F1 ,→ · · · ,→ Fm = F.

We say that F is a holomorphic filtration if the sheaves Fi are locally free. A G–equivariant morphism from a G–equivariant sheaf filtration (2.9)

0 F 0 : 0 ,→ F00 ,→ F10 ,→ · · · ,→ Fm = F 0,

to another one F given by (2.8) is a G–equivariant morphism f : F 0 → F, with f (Fi0 ) = Fi ∩ Im(f ) for each i. Let p : X × P1 → X and q : X × P1 → P1 be the canonical projections. Unless otherwise stated, by an SL(2, C)–equivariant sheaf filtration on X × P1 we shall mean a SL(2, C)–equivariant sheaf filtration on X × P1 , such that there are isomorphisms of SL(2, C)–sheaves (2.10) Fi /Fi−1 ∼ = p∗ Ei ⊗ q ∗ O(2i), for 1 ≤ i ≤ m. Theorem 2.1. 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. Proof. We prove it first for holomorphic SL(2, C)-equivariant vector bundles on X ×P1 . As we explained before, there is an equivalence between SL(2, C)-equivariant holomorphic vector bundles H on X × P1 and P -equivariant holomorphic vector bundles G on X. The C∗ -action on any such G is given by a weight decomposition (2.11) G = ⊕λ∈∆(G) Gλ ⊗Mλ , Gλ = HomC∗ (Mλ , G) for λ ∈ Z, ∆(G) = {λ ∈ Z|Gλ 6= 0}, where Mλ is now the irreducible C∗ -equivariant vector bundle over X associated to the irreducible representation Mλ (we use the same letter for both), for λ ∈ Z, and the set of weights ∆(G) is finite because G has finite rank. The key ingredient to prove the previous decomposition is the existence of a (unique) invariant holomorphic

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projection operator Π : G → G onto the C∗ -invariant part of E. This is proved e.g. in ∗ [AB]. Using the familiar notation (−)C := Π(−), the multiplicity bundles are then ∗ Gλ = HomC∗ (Mλ , G) = Hom(Mλ , G)C , where the bundle Hom(Mλ , G) has the canonical C∗ –action. As we saw before, the rest of the P –action on G, given by the action of X − , is specified by a collection of morphisms ϕλ : Gλ → Gλ−2 , and if G is indecomposable, then ∆(G) is a finite sequence m− , m− + 2, . . . , m+ − 2, m+ of consecutive odd or even integers. Twisting E := G ⊗ M−m− , we translate the weights into (2.12) E = ⊕m i=0 Ei ⊗ M2i ,

Ei := Gm− +2i , ∆(E) = {0, 2, . . . , 2m}, m := (m+ − m− )/2.

The X − -action, given by morphisms φi : Ei → Ei−1 , defines the diagram (2.7). As in (2.5), this holomorphic chain defines a filtration of P –invariant holomorphic vector subbundles E (≤•) : 0 ⊂ E(≤0) ⊂ E(≤1) ⊂ . . . ⊂ E(≤m) , (2.13) E(≤i) /E(≤i−1) ∼ = Ei ⊗ M2i , 0 ≤ i ≤ m. Defining Fi = SL(2, C) ×P E(≤i) , and noting that the homogeneous bundle infuced by Mλ on P1 is O(λ) = SL(2, C) ×P Mλ , we obtain the filtration F : 0 ,→ F0 ,→ F1 ,→ · · · ,→ Fm = F, (2.14) Fi /Fi−1 ∼ = p∗ Ei ⊗ q ∗ O(2i), 0 ≤ i ≤ m. Finally, if H := SL(2, C) ×P G then F := SL(2, C) ×P E is given by F = H×q ∗ O(−m− ). To prove the theorem for general coherent SL(2, C)-equivariant sheaves on X × P1 , we combine the arguments given for holomorphic SL(2, C)-equivariant vector bundles on X × P1 with the lemmas below. Lemma 2.2. There is an equivalence between the category of coherent SL(2, C)-equivariant sheaves on X × P1 , and the category of coherent P -equivariant sheaves on X. Lemma 2.3. A coherent SL(2, C)-equivariant sheaf G on X admits a weight decomposition (2.15) E∼ = ⊕λ∈∆(E) Eλ ⊗ Mλ , where ∆(G) ⊂ Z is finite, and Eλ is a coherent sheaf with trivial C∗ –action, for each λ ∈ ∆(G). To prove these lemmas, we need the following definition. Let G be a complex Lie group and M a complex G–manifold such that the quotient M/G is a complex manifold. We define a functor F 7→ F G from the category of coherent G-equivariant sheaves on M to the category of (not necessarily coherent) sheaves on M/G. Given a coherent Gequivariant sheaf F on M , and U ⊂ M/G open, by definition F G (U ) := F(π −1 (U ))G , where π : M → M/G is projection. Proof of Lemma 2.2. Let E be a coherent P -equivariant sheaf on X. The P -action on SL(2, C), given by p · g = gp−1 , for p ∈ P , g ∈ SL(2, C), induces a P -action on the structure sheaf OSL of SL(2, C), hence on OM , where M = X × SL(2, C). Let πX : M → X and πSL : M → SL(2, C) be projections. The coherent sheaf ∗ ∗ H = πX E ⊗ πSL OSL on M has a holomorphic P -equivariant action. By definition, the SL(2, C)-equivariant sheaf induced by E on X × P1 is its P -invariant part F = HP . To prove that it is coherent, one sees that the set of points y ∈ X × P1 where F does

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not admit a presentation by locally free sheaves, is S(F) = S(E) × P1 , where S(E) is the set of points x ∈ X where E does not admit a presentation by locally free sheaves. But S(E) is empty by hypothesis, so S(F) is also empty.  Proof of Lemma 2.3. Firstly, for any coherent C∗ –equivariant sheaf on X (the C∗ action on X is trivial), and for each x ∈ X, there is a neighbourhood U of x, finite dimensional complex representations V, W of C∗ , and a C∗ –equivariant exact sequence (2.16)

g

f

OB ⊗ W −−−→ OB ⊗ V −−−→ E|B −−−→ 0,

such that, for any U 0 ⊂ U open, each s ∈ E(U 0 ) is C∗ -finite, i.e. the C-linear span of the orbit L · s is finite-dimensional. This result is an easy consequence of [R, Proposition 2.1]. Secondly, we show that for each coherent C∗ -equivariant sheaf E on X, there is ∗ a unique C∗ -invariant projection operator Π : E → E onto E C , and that it commutes with restrictions of sections of the sheaf E, as well as with (C∗ -equivariant) morphisms of coherent C∗ -equivariant sheaves on X. This is an immediate consequence of [R, Proposition 2.2]. The last result that we need is that the functor which takes a coherent ∗ C∗ -equivariant sheaf E on X into its invariant part E C is exact (this is standard in representation theory of Lie groups; in our case it follows from the existence of Π), and ∗ ∗ that E C is coherent. The coherence of E C follows from the exactness of the previous functor and from the existence of C∗ –equivariant exact sequences (2.16). The isotopical decompositions now follows from the fact that each section in E(U 0 ) is C∗ -finite, and from the existence of the invariant projection Π.  3. Hitchin–Kobayashi correspondence and dimensional reduction Let F be a holomorphic vector bundle over a compact n-dimensional K¨ahler manifold (M, ω), whose volume is normalised to 2π. A hermitian metric h on F defines a unique unitary connection Ah —the so-called Chern connection— on (F, h) whose associated ¯ ∂-operator is compatible with the holomorphic structure of F; let Fh be its curvature. We say that h is Hermite–Einstein if it satisfies √ (3.17) −1 ΛFh = µ(F)I, 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 0 , is defined as µ(F 0 ) =

deg F 0 . rk F 0

Here

ω n−1 (n − 1)! M 0 0 is the degree of F , where c1 (F ) is the first Chern class of F 0 , and rk F 0 is the rank of F 0. 0

deg F =

Z

c1 (F 0 ) ∧

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 stable if for every proper coherent subsheaf F 0 ,→ F, µ(F 0 ) < µ(F), where proper means that 0 < rk F 0 < rk F. A direct sum of stable holomorphic vector bundles, all of them with the same slope, is called polystable.

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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 [NS] (also proved using gaugetheoretic methods in [D1]), and for higher dimensions is the Theorem of Donaldson, Uhlenbeck and Yau [D2, D3, UY] (see also [K]). Let F be a holomorphic filtration over a compact K¨ahler manifold (M, ω) given by (3.18)

F : 0 ,→ F0 ,→ F1 ,→ · · · ,→ Fm = F.

The deformed Hermite–Einstein equation for a hermitian metric on F involves as many parameters as steps are in the filtration, and has the form   τ0 I0   √ τ1 I1 , (3.19) −1 ΛFh =  .   .. τm Im

where Fh is the curvature of the Chern connection of h, and the RHS is a diagonal matrix, with constants τ0 , τ1 , . . . , τm ∈ R, written in blocks corresponding to the splitting which h defines in the filtration F . Taking traces in (3.19) and integrating over M , we see that the parameters are constrained by (3.20)

m X

τi rk(Fi /Fi−1 ) = deg F,

i=0

which means that there are only m independent parameters, as one would expect. If τ0 = · · · = τm = µ(F), (3.19) reduces of course to the Hermite–Einstein equation (3.17). A metric h on F satisfying (3.19) is called a τ -Hermite–Einstein. 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. Let F , as in (3.18), be a holomorphic filtration on M . A sheaf subfiltration of F is a sheaf filtration (3.21)

0 F 0 : 0 ,→ F00 ,→ F10 ,→ · · · ,→ Fm = F 0,

where F 0 is a subsheaf of F, such that Fi0 = Fi ∩ F 0 for 0 ≤ i ≤ m. The sheaf subfiltration F 0 ,→ F is called proper if 0 < rk F 0 < rk F. Let α = (α0 , . . . , αm−1 ) ∈ Rm , and let F 0 be a sheaf filtration of F . We define its α-degree and α-slope respectively by m−1 X degα F 0 0 0 degα F = deg F + αi rk(Fi0 ), µα (F 0 ) = . rk F 0 i=0 We say that the sheaf filtration F is α-stable if for all proper sheaf subfiltrations F 0 ,→ F , we have µα (F 0 ) < µα (F ). A α-polystable sheaf filtration is a direct sum of α-stable sheaf filtrations, all of them with the same α-slope. One has the following Hitchin–Kobayashi correspondence ([AG, Theorem 2.1]. Theorem 3.1. Let F be a holomorphic filtration on M . Let τ = (τ0 , . . . , τm ) ∈ Rm+1 satisfy (3.20), and let α = (α0 , . . . , αm−1 ) ∈ Rm be an m-tuple of positive real numbers

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defined by (3.22)

αi = τi+1 − τi

(0 ≤ i ≤ m − 1).

Then F admits a τ -Hermite–Einstein metric if and only if it is α-polystable. One has an equivariant version of this theorem for the SL(2, C)-manifold M = X ×P1 , by considering SL(2, C)-invariant sheaf subfiltrations of F and SU(2)-invariant hermitian metrics on F ([AG, Theorem 2.3]). Theorem 3.2. Let F be a SL(2, C)-equivariant holomorphic filtration on X × P1 . Let τ = (τ0 , . . . , τm ) ∈ Rm+1 satisfy (3.20), and let α = (α0 , . . . , αm−1 ) ∈ Rm be an m-tuple of positive real numbers given by (3.22). Then F is SL(2, C)-invariantly α-polystable if and only if it admits SU(2)-invariant τ -Hermite–Einstein metric. By means of gauge-theoretic dimensional reduction methods we can analyse SU(2)invariant solutions to the deformed Hermitian-Einstein equation for a filtration on X × P1 to obtain some equations on X. To be more precise, let F be an SL(2, C)equivariant holomorphic filtration on X × P1 and let C = (E, φ) be its corresponding 0 holomorphic chain over X (given by (2.7). Let τ = (τ0 , . . . , τm ), τ 0 = (τ00 , . . . , τm )∈ m+1 R be related by τi0 = τi + 4πi (0 ≤ i ≤ m). Then one can prove [AG, Theorem 4.1] that F admits an SU(2)-invariant τ 0 -Hermite– Einstein metric if and only if C admits an (m + 1)-tuple h = (h0 , h1 , . . . , hm ) of hermitian metrics where hi is a metric on Ei satisfying √ −1ΛFh0 + 12 φ1 ◦ φ∗1 = τ0 idE0 , √ 1 ∗ ∗ (3.23) −1ΛF (1 ≤ i ≤ m − 1), √ hi − 2 (φi ◦ 1φi ∗− φi+1 ◦ φi+1 ) = τi idEi , −1ΛFhm − 2 φm ◦ φm = τm idEm . These are called the chain τ -vortex equations, and generalize the vortex equations studied in [G, BG]. Let C = (E, φ) a holomorphic chain, as in (2.7). A sheaf subchain of C is a sheaf chain (3.24)

φ0

φ0m−1

φ0

m 1 0 0 C 0 : Em −→ Em−1 −→ · · · −→ E00 ,

such that Ei0 is a subsheaf of Ei for 0 ≤ i ≤ m, and φi ◦ fi = fi−1 ◦ φ0i for 1 ≤ i ≤ m, where fi : Ei0 ,→ Ei are the inclusion morphisms. The sheaf subchain C 0 ,→ C is called Pm Pm 0 proper if 0 < i=0 rk Ei < i=0 rk Ei . Let α = (α0 , α1 , . . . , αm ) ∈ Rm . The α-degree and α-slope of a coherent sheaf chain C 0 , are defined by m m X X deg C 0 0 0 degα C = deg Ei − αi rk Ei0 , µα (C 0 ) = Pm α 0 , i=0 rk Ei i=0 i=0

respectively. We say that the coherent sheaf chain C is α-stable if for all proper sheaf subchains C 0 ,→ C, µα (C 0 ) < µα (C). A direct sum of α-stable coherent sheaf chains, all of them with the same α-slope, is called α-polystable. Using the correspondence proved in the previous section, one can show a dimensional reduction reduction result relating filtrations to chains. More precisely, let F be an SL(2, C)-equivariant holomorphic filtration on X × P1 , and let C be the corresponding

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0 holomorphic chain on X. Let α = (α0 , . . . , αm−1 ) ∈ Rm , α0 = (α00 , . . . , αm ) ∈ Rm+1 be related by i−1 X α0 = 0, αi = αj0 − 4πi (1 ≤ i ≤ m). j=0

Then F is SL(2, C)-invariantly α0 -stable if and only if C is α-stable. Combining all these results we have the following Hitchin–Kobayashi correspondence [AG, Theorem 3.1] generalizing that obtained for triples in [BG]. Theorem 3.3. 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. References [Ak] [AB] [AG] [BG] [D1] [D2] [D3] [G] [H] [K] [NS] [R] [UY]

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Department of Mathematics, University of Illinois at Urbana-Champaign, IL 61801,USA E-mail address: [email protected] ´ticas, Universidad Auto ´ noma de Madrid, 28049 Madrid, Departamento de Matema SPAIN E-mail address: [email protected]