HITCHIN KOBAYASHI CORRESPONDENCE, QUIVERS, AND VORTICES INTRODUCTION

Transcription

1 ËÁ Ì ÖÛÒ ËÖĐÓÒÖ ÁÒØÖÒØÓÒÐ ÁÒ ØØÙØ ÓÖ ÅØÑØÐ ÈÝ ÓÐØÞÑÒÒ ¹½¼¼ ÏÒ Ù ØÖ ÀØÒßÃÓÝ ÓÖÖ ÓÒÒ ÉÙÚÖ Ò ÎÓÖØ ÄÙ ÐÚÖÞßÓÒ ÙÐ Ç Ö ÖßÈÖ ÎÒÒ ÈÖÖÒØ ËÁ ½¾ ¾¼¼ µ ÂÒÙÖÝ ½ ¾¼¼ ËÙÓÖØ Ý Ø Ù ØÖÒ ÖÐ ÅÒ ØÖÝ Ó ÙØÓÒ ËÒ Ò ÙÐØÙÖ ÚÐÐ Ú ØØ»»ÛÛÛº ººØ

2 HITCHIN KOBAYASHI CORRESPONDENCE, QUIVERS, AND VORTICES LUIS ÁLVAREZ CÓNSUL AND OSCAR GARCÍA PRADA ABSTRACT. A twisted quiver bundle is a set of holomorphic vector bundles over a complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles, labelled by the arrows. When the manifold is Kähler, quiver bundles admit natural gauge-theoretic equations, which unify many known equations for bundles with extra structure. In this paper we prove a Hitchin Kobayashi correspondence for twisted quiver bundles over a compact Kähler manifold, relating the existence of solutions to the gauge equations to a stability criterion, and consider its application to a number of situations related to Higgs bundles and dimensional reductions of the Hermitian Einstein equations. INTRODUCTION A quiver É consists of a set É ¼ of vertices Ú Ú ¼, and a set É ½ of arrows Ú Ú ¼ connecting the vertices. Given a quiver and a compact Kähler manifold, a quiver bundle is defined by assigning a holomorphic vector bundle Ú to a finite number of vertices and a homomorphism Ú Ú ¼ to a finite number of arrows. A quiver sheaf is defined by replacing the term holomorphic vector bundle by coherent sheaf in this definition. If we fix a collection of holomorphic vector bundles Å parametrized by the set of arrows, and the morphisms are Ú Å Å Ú ¼, twisted by the corresponding bundles, we have a twisted quiver bundle or a twisted quiver sheaf. In this paper we define natural gauge-theoretic equations, that we call quiver vortex equations, for a collection of hermitian metrics on the bundles associated to the vertices of a twisted quiver bundle (for this, we need to fix hermitian metrics on the twisting vector bundles). To solve these equations, we introduce a stability criterion for twisted quiver sheaves, and prove a Hitchin Kobayashi correspondence, relating the existence of (unique) hermitian metrics satisfying the quiver vortex equations to the stability of the quiver bundle. The equations and the stability criterion depend on some real numbers, the stability parameters (cf. Remarks 2.6 for the exact number of parameters). It is relevant to point out that our results cannot be derived from the general Hitchin Kobayashi correspondence scheme developed by Banfield [Ba] and further generalized by Mundet [M]. This is due not only to the presence of twisting vector bundles, but also to the deformation of the Hermitian Einstein terms in the equations. This deformation is naturally explained by the symplectic interpretation of the equations, and accounts for extra parameters in the stability condition for the twisted quiver bundle. This correspondence provides a unifying framework to study a number of problems that have been considered previously. The simplest situation occurs when the quiver has a single vertex and no arrows, in which case a quiver bundle is just a holomorphic bundle, and the gauge equation is the Hermitian Einstein equation. A theorem of Donaldson, Uhlenbeck and Yau [D1, D2, UY], establishes that a (unique) solution to the Hermitian Einstein equation exists if and only if is polystable. The bundle is called stable (in the sense of Mumford Takemoto) if µ µ for each proper coherent subsheaf, where the slope µ is the degree divided by the rank; a finite direct sum of stable bundles with the same slope is called polystable. A correspondence of this type is usually known as a Hitchin Kobayashi correspondence. A Hitchin Kobayashi correspondence, where some extra structure is added to the bundle, appears in the theory of Higgs bundles, consisting of pairs Date: 10 December 2001; in revised form, 16 December Mathematics Subject Classification. Primary: 58C25; Secondary: 58A30, 53C12, 53C55, 83C05. 1

3 2 LUIS ÁLVAREZ CÓNSUL AND OSCAR GARCÍA PRADA µ formed by a holomorphic vector bundle and a morphism Å Å, where Å is the sheaf of holomorphic differentials (sometimes the condition ¼ is added as part of the definition). Higgs bundles were first studied by Hitchin [H] (when is a compact Riemann surface), and Simpson [S] (when is higher dimensional), who introduced a natural gauge equation for them, and proved a Hitchin Kobayashi correspondence. Higgs bundles are twisted quiver bundles, for a quiver formed by one vertex and one arrow whose head and tail coincide, and the twisting bundle is the holomorphic tangent bundle (i.e. the dual to Å). Another class of quiver bundles are holomorphic triples ½ ¾ µ, consisting of two holomorphic bundles ½ and ¾, and a morphism ¾ ½. The quiver has two vertices, say ½ and ¾, and one arrow ¾ ½ (the twisting sheaf is Ç ). The corresponding equations are called the coupled vortex equations [G2, BG]. When ¾ Ç, holomorphic triples are holomorphic pairs µ, where is a bundle and ¾ À ¼ µ (cf. [B]). There are other examples of quiver vortex equations that come out naturally from the study of the moduli of solutions to the Higgs bundle equation. Combining a theorem of Donaldson and Corlette [D3, C] with the Hitchin Kobayashi correspondence for Higgs bundles [H, S], one has that the set of isomorphism classes of semisimple complex representations of the fundamental group of in Ä Ö µ is in bijection with the moduli space of polystable Higgs bundles with vanishing Chern classes. When is a compact Riemann surface, this generalizes a theorem of Narasimhan and Seshadri [NS], which provides an interpretation of the unitary representations of the fundamental group as degree zero polystable vector bundles, up to isomorphism. Now, if is a compact Riemann surface of genus ¾, the Morse methods introduced by Hitchin [H] reduce the study of the topology of the moduli Å of Higgs bundles to the study of the topology of the moduli of complex variations of the Hodge structure the critical points of the Morse function in this case. These are twisted quiver bundles, called twisted holomorphic chains, for a quiver whose vertex set is the set of integer numbers, and whose arrows are ½, for each ¾ ; the twisting bundle associated to each arrow is the holomorphic tangent bundle. The twisted holomorphic chains that appear in these critical submanifolds are polystable for particular values of the stability parameters. Using Morse theory, Hitchin [H] computed the Poincaré polynomial of Å for the rank 2 case. Gothen [Go] obtained similar results for rank 3: the critical submanifolds are moduli spaces of stable twisted holomorphic chains formed by a line bundle and a rank 2 bundle (i.e. twisted holomorphic triples), and by three line bundles. To use these methods for higher rank, one needs to study moduli spaces of other twisted holomorphic chains. A possible strategy is to proceed as in [Th], studying the moduli of twisted holomorphic chains in the whole parameter space. Another interesting type of quiver bundles arise in the study of semisimple representations of the fundamental group of in Í Õµ, the unitary group for a hermitian inner product of indefinite signature. Here, the quiver has two vertices, say ½ and ¾, and two arrows, ½ ¾ and ¾ ½, and the twisting bundle associated to each arrow is the holomorphic tangent bundle. These are studied in [BGG1, BGG2]. Another context in which quiver bundles appear naturally is in the study of dimensional reductions of the Hermitian Einstein equation over the product of a Kähler manifold and a flag manifold. In this case, the parabolic subgroup defining the flag manifold entirely determines the structure of the quiver [AG1, AG2]. The dimensional reduction for this kind of manifolds has provided insight in the general theory of quiver bundles, and was actually the first method used to prove a Hitchin Kobayashi correspondence for holomorphic triples [G2, BG], holomorphic chains [AG1], and quiver bundles for more general quivers with relations [AG2]. In these examples, the quiver bundles are not twisted, however, there are other examples for which a generalization of the method of dimensional reduction has produced twisted holomorphic triples [BGK1, BGK2]. An important feature of the stability of quiver sheaves is that it generally depends on several real parameters. When is an algebraic variety, the ranks and degrees appearing in the numerical condition defining the stability criterion are integral, and the parameter space is partitioned into chambers. Strictly semistable quiver sheaves can occur when the parameters are on a wall separating the chambers, and the stability condition only depends on the chamber in which the parameters are. In the

4 HITCHIN KOBAYASHI CORRESPONDENCE, QUIVERS, AND VORTICES 3 case of holomorphic triples [BG], there is a chamber (actually an interval in Ê) where the stability of the triple is related to the stability of the bundles. This can be used to obtain existence theorems for stable triples when the parameters are in this chamber, while the methods of [Th] can be used to prove existence results for other chambers (see [BGG2] for recent work in the case of triples). The geography of the resulting convex polytope for other quivers is an interesting issue to which we wish to return in a future paper. To approach this problem, one should study the homological algebra of quiver bundles. This has been developed by Gothen and King in a paper [GK] that appeared after we submitted this paper. When the manifold is a point, a quiver bundle is just a quiver module (over ; cf. e.g. [ARS]). For arbitrary, a quiver bundle can be regarded as a family of quiver modules (the fibres of the quiver bundle), parametrized by. One can thus transfer to our setting many constructions of the theory of quiver modules. In the last part of the paper we introduce a more algebraic point of view by considering the path algebra bundle of the twisted quiver and looking at twisted quiver bundles as locally free modules over this bundle of algebras. This point of view is inspired by a similar construction for quiver modules [ARS], and suggests a generalization to other algebras that appear naturally in other problems. This is something to which we plan to come back in the future. The Hitchin Kobayashi correspondence for quiver bundles combines in one theory two different versions, in some sense, of the theorem of Kempf and Ness [KN] identifying the symplectic quotient of a projective variety by a compact Lie group action, with the geometric invariant theory quotient. The first one is the classical Hitchin Kobayashi correspondence for vector bundles, and the second one occurs when the manifold is a point, in which case the equations and the stability condition reduce to the moment map equations and the stability condition for quiver modules introduced by King [K]. As we prove in Theorem 4.3, there is in fact a very tight relation between the quiver vortex equations and the moment map equations for quiver modules: when the twisting sheaves are Ç and the bundles have vanishing Chern classes, the existence of solutions to the quiver vortex equations is equivalent to the existence of flat metrics on the bundles which fibrewise satisfy the moment map equations for quiver modules. 1. TWISTED QUIVER BUNDLES In this section we define the basic objects that we shall study: twisted quiver bundles and twisted quiver sheaves. They are representations of quivers in the categories of holomorphic vector bundles and coherent sheaves, respectively, twisted by some fixed holomorphic vector bundles, as explained in Ü1.2. Thus, many results about quiver modules, i.e. quiver representations in the category of vector spaces, can be tranferred to our setting. A good reference for quivers and their linear representations is [ARS] Quivers. A quiver, or directed graph, is a pair of sets É É ¼ É ½ µ together with two maps Ø É ½ É ¼. The elements of É ¼ (resp. É ½ ) are called the vertices (resp. arrows) of the quiver. For each arrow ¾ É ½, the vertex Ø (resp. ) is called the tail (resp. head) of the arrow. The arrow is sometimes represented by Ú Ú ¼ when Ú Ø and Ú ¼ Twisted quiver sheaves and bundles. Throughout this paper, is a connected compact Kähler manifold, É is a quiver, and Å is a collection of finite rank locally free sheaves Å on, for each arrow ¾ É ½. By a sheaf on, we shall will mean an analytic sheaf of Ç -modules. Our basic objects are given by the following: Definition 1.1. An Å-twisted É-sheaf on is a pair Ê µ, where is a collection of coherent sheaves Ú on, for each Ú ¾ É ¼, and is a collection of morphisms Ø Å Å, for each ¾ É ½, such that Ú ¼ for all but finitely many Ú ¾ É ¼, and ¼ for all but finitely many ¾ É ½.

5 4 LUIS ÁLVAREZ CÓNSUL AND OSCAR GARCÍA PRADA Remark 1.2. Given a quiver É É ¼ É ½ µ, as defined in Ü1.1, the sets É ¼ and É ½ can be infinite, but for each Å-twisted É-sheaf Ê µ, the subset É ¼ ¼ É ¼ of vertices Ú such that Ú ¼, and the subset É ¼ ½ É ½ of arrows such that ¼, are both finite. Thus, to any Å-twisted É-sheaf Ê µ, we can associate the subquiver É ¼ É ¼ ¼ ɼ ½ µ of É, and Ê can be seen as an Å ¼ -twisted É ¼ -sheaf, where É ¼ ¼, ɼ ½ are finite sets, and Å ¼ Å is the collection of sheaves Å with ¾ É ¼ ½. As usual, we identify a holomorphic vector bundle, with the locally free sheaf of sections of. Accordingly, a holomorphic Å-twisted É-bundle is an Å-twisted É-sheaf Ê µ such that the sheaf Ú is a holomorphic vector bundle, for each Ú ¾ É ¼. For the sake of brevity, in the following the terms É-sheaf or É-bundle are to be understood as Å-twisted É-sheaf or Å-twisted É- bundle, respectively, often suppressing the adjective Å-twisted. A morphism Ê Ê ¼ between two É-sheaves Ê µ, Ê ¼ ¼ ¼ µ, is given by a collection of morphisms Ú Ú Ú, ¼ for each Ú ¾ É ¼, such that ¼ Æ Ú Å Å µ Ú ¼ Æ, for each arrow Ú Ú ¼ in É. If Ê Ê ¼ and Ê ¼ Ê ¼¼ are two morphisms between representations Ê µ, Ê ¼ ¼ ¼ µ, Ê ¼¼ ¼¼ ¼¼ µ, then the composition Æ is defined as the collection of composed morphisms Ú Æ Ú Ú ¼¼ Ú, for each Ú ¾ É ¼. We have thus defined the category of Å-twisted É-sheaves on, which is abelian. Important concepts in relation to stability and semistability (defined in Ü2.3) are the notions of É-subsheaves and quotient É-sheaves, as well as indecomposable and simple É-sheaves. They are defined as for any abelian category. In particular, an Å-twisted É-subsheaf of Ê µ is another Å-twisted É-sheaf Ê ¼ ¼ ¼ µ such that Ú ¼ Ú, for each Ú ¾ É ¼, Ø ¼ Å Å µ ¼, for each ¾ É ½, and ¼ ¼ Ø Å Å ¼ is the restriction of to Ø ¼ Å Å, for each ¾ É ¼. 2. GAUGE EQUATIONS AND STABILITY 2.1. Gauge equations. Throughout this paper, given a smooth bundle on, Å µ (resp. Å µ) is the space of smooth -valued complex -forms (resp. µ-forms) on, is a fixed Kähler form on, and Å µ Å ½ ½ µ is contraction with (we use the same notation as e.g. in [D1]). The gauge equations will also depend on a fixed collection Õ of hermitian metrics Õ on Å, for each ¾ É ½, which we fix once and for all. Let Ê µ be a holomorphic Å-twisted É-bundle on. A hermitian metric on Ê is a collection À of hermitian metrics À Ú on Ú, for each Ú ¾ É ¼ with Ú ¼. To define the gauge equations on Ê, we note that Ø Å Å has a smooth adjoint morphism À Ø Å Å with respect to the hermitian metrics À Ø Å Õ on Ø Å Å, and À on, for each ¾ É ¼, so it makes sense to consider the composition Æ À Ø Å Å. Moreover, and À can be seen as morphisms Ø Å Å and À Å Å Ø, so À Æ Ø Ø makes sense too. Definition 2.1. Let and be collections of real numbers Ú Ú, with Ú positive, for each Ú ¾ É ¼. A hermitian metric À satisfies the Å-twisted quiver µ-vortex equations if (2.2) Ú ½ ÀÚ Æ Ú Ú ¾ ½ Úµ Æ À ¾Ø ½ Úµ for each Ú ¾ É ¼ such that Ú ¼, where ÀÚ is the curvature of the Chern connection ÀÚ associated to the metric À Ú on the holomorphic vector bundle Ú, for each Ú ¾ É ¼ with Ú ¼ Moment map interpretation. The twisted quiver vortex equations appear as a symplectic reduction condition, as we explain now. Let be a collection of smooth vector bundles Ú, for each Ú ¾ É ¼, with Ú ¼ for all but finitely many Ú ¾ É ¼. By removing the vertices Ú ¾ É ¼ with Ú ¼ and all but finitely many arrows ¾ É ½, we obtain a finite subquiver, which we still call É É ¼ É ½ µ, such that Ú ¼ for each Ú ¾ É ¼ (see Remark 1.2). Let À Ú be a hermitian metric on Ú, for each Ú ¾ É ¼. Let Ú and Ú be the corresponding spaces of unitary connections and their À

6 HITCHIN KOBAYASHI CORRESPONDENCE, QUIVERS, AND VORTICES 5 unitary gauge groups, and let Ú ½½ Ú be the space of unitary connections Ú with Ú µ ¾ ¼, for each Ú ¾ É ¼. The group Ú acts on the space of unitary connections, and on the representation space Å ¼, defined by (2.3) Ú Å ¼ Å ¼ Ê É µµ ÛØ Ê É µ Å ¾É ½ ÀÓÑ Ø Å Å µ where ÀÓÑ Ø Å Å µ is the vector bundle of homomorphisms Ø Å Å. An element ¾ is a collection of group elements Ú ¾ Ú, for each Ú ¾ É ¼, and an element ¾ (resp. ¾ Å ¼ ) is a collection of unitary connections Ú ¾ Ú (resp. smooth morphisms Ø Å Å ), for each Ú ¾ É ¼ (resp. ¾ É ½ ). The -actions on and Å ¼ are µ ¼ ÛØ ¼ Ú Ú Æ Ú Æ ½ Ú Æ Æ ½ Ø ÓÖ Ú ¾ É ¼ Å ¼ Å ¼ µ ¼ ÛØ ¼ Å Å µ ÓÖ ¾ É ½ respectively. The induced -action on the product Å ¼ leaves invariant the subset Æ of pairs µ such that Ú ¾ ½½ Ú, for each Ú ¾ É ¼, and Ø Å Å is holomorphic with respect to Ø and, for each ¾ É ¼. Let Ú be the Ú-invariant symplectic form on Ú, for each Ú ¾ É ¼, as given in [AB] for a compact Riemann surface, or e.g. in [DK, Proposition 6.5.8] for any compact Kähler manifold, that is, Ú Ú Ú µ ØÖ Ú Ú µ ÓÖ Ú Ú ¾ Å ½ Ú µµ where Ú µ is the vector bundle of À Ú -antiselfadjoint endomorphisms of Ú. The corresponding moment map Ú Ú Ä Úµ is given by Ú Ú µ Ú (we use implicitly the inclusion of Ä Ú in its dual space by means of the metric À Ú on Ú ). The symplectic form Ê on Å ¼ associated to the Ä ¾ -metric induced by the hermitian metrics on the spaces Å ¼ ÀÓÑ Ø Å Å µµ is - invariant, and has associated moment map Ê Å ¼ È Ä µ given by Ê ÊÚ, with ÊÚ Å ¼ Ä Ú Ä Ä µ given by (2.4) ½ ÊÚ µ ¾ ½ Úµ Æ À ¾Ø ½ Úµ À Æ ÓÖ ¾ Å ¼ (this follows as in [K, Ü6], which considers the action of a unitary group on a representation space of quiver modules). Given a collection of real numbers Ú ¼, for each Ú ¾ É ¼, È Ú Ú Ê is obviously a -invariant symplectic form on Å ¼. A moment map for this symplectic form is È Ú Ú Ê, where we are omitting pull-backs to Å ¼ in the notation. Any collection of real numbers Ú, for each Ú ¾ É ¼ defines an element ½ ½ È Ú Ú in the center of Ä. The points of the symplectic reduction ½ ½ µ are precisely the orbits of pairs µ such that the hermitian metric À satisfies the Å-twisted µ-vortex quiver equations on the corresponding holomorphic quiver bundle Ê µ. Thus, Definition 2.1 picks up the points of ½ ½ µ in the Kähler submanifold (outside its singularities) Æ. For convenience in the Hitchin Kobayashi correspondence, it is formulated in terms of hermitian metrics Stability. To define stability, we need some preliminaries and notation. Let Ò be the complex dimension of. Given a torsion-free coherent sheaf on, the double dual sheaf Ø µ is a holomorphic line bundle, and we define the first Chern class ½ µ of as the first Chern class of Ø µ. The degree of is the real number ¾ ½ µ ÎÓÐ µ Ò ½µ Å ½ µ Ò ½ «where ÎÓÐ µ is the volume of, Ò ½ is the cohomology class of Ò ½, and is the fundamental class of. Note that the degree depends on the cohomology class of. Given a holomorphic

7 6 LUIS ÁLVAREZ CÓNSUL AND OSCAR GARCÍA PRADA vector bundle on, by Chern-Weil theory, its degree equals ½ µ ØÖ ½ À µ ÎÓÐ µ where À is the curvature of the Chern connection associated to a hermitian metric À on. Let É be a quiver, and, be collections of real numbers Ú Ú, with Ú ¼, for each Ú ¾ É ¼ ; and are called the stability parameters. Let Ê µ be a É-sheaf on. Definition 2.5. The µ-degree and µ-slope of Ê are ʵ Ú Ú µ Ú Ö Ú µµ ʵ ʵ È Ú Ö Ú µ respectively. The É-sheaf Ê is called µ-stable (resp. µ-semistable) if for all proper É- subsheaves Ê ¼ of Ê, Ê ¼ µ ʵ (resp. Ê ¼ µ ʵ). A µ-polystable É-sheaf is a direct sum of µ-stable É-sheaves, all of them with the same µ-slope. As for coherent sheaves, one can prove that any µ-stable É-sheaf is simple, i.e. its only endomorphisms are the multiples of the identity. Remarks 2.6. (i) If a holomorphic É-bundle Ê admits a hermitian metric satisfying the µ- vortex equations, then taking traces in (2.2), summing for Ú ¾ É ¼, and integrating over, we see that the parameters are constrained by ʵ ¼. (ii) If we transform the parameters, multiplying by a global constant ¼, obtaining ¼, ¼, then ¼ ¼ ʵ ʵ. Furthermore, if we transform the parameters by ¼ Ú Ú Ú for some ¾ Ê, and let ¼, then ¼ ¼ ʵ ʵ. Since the stability condition does not change under these two kinds of transformations, the effective number of stability parameters of a quiver sheaf Ê µ is ¾Æ ʵ ¾, where Æ Êµ ¼. From the point of view of the is the (finite) number of vertices Ú ¾ É ¼ with Ú vortex equations (2.2), the first type of transformations, ¼, ¼, corresponds to a redefinition of the sections ¼ ½¾ (note that the stability condition is invariant under this transformation), while the second type corresponds to the constraint ʵ ¼ in (i). (iii) As usual with stability criteria, in Definition 2.5, to check µ-stability of a É-sheaf Ê, it suffices to check Ê ¼ µ ʵ for the proper É-subsheaves Ê ¼ Ê such that Ú ¼ Ú is saturated, i.e. such that the quotient Ú Ú ¼ is torsion-free, for each Ú ¾ É ¼. 3. HITCHIN KOBAYASHI CORRESPONDENCE In this section we will prove a Hitchin Kobayashi correspondence between the twisted quiver vortex equations and the stability condition for holomorphic twisted quiver bundles: Theorem 3.1. Let and be collections of real numbers Ú and Ú, respectively, with Ú ¼, for each Ú ¾ É ¼. Let Ê µ be a holomorphic Å-twisted É-bundle such that ʵ ¼. Then Ê is µ-polystable if and only if it admits a hermitian metric À satisfying the quiver µ-vortex equations (2.2). This hermitian metric À is unique up to an automorphism of the É-bundle, i.e. up to a multiplication by a constant ¼ for each µ-stable summand Ê of Ê Ê ½ Ê Ð. Remark 3.2. This theorem generalizes previous theorems, mainly Donaldson Uhlenbeck Yau theorem [D1, D2, UY], the Hitchin Kobayashi correspondence for Higgs bundles [H, S], holomorphic triples and chains [AG1, BG], twisted holomorphic triples [BGK2], etc. It should be mentioned that Theorem 3.1 does not follow from the general theorems proved in [Ba, M] for the following two reasons. First, the symplectic form È Ú Ú Ê on Å ¼ (cf. Ü2.2) has been deformed by the parameters whenever Ú Ú ¼ for some Ú Ú ¼ ¾ É ¼ ; as a matter of fact, the vortex equations (2.2) depend on new parameters even for holomorphic triples or chains [AG1, BG], hence generalizing

8 HITCHIN KOBAYASHI CORRESPONDENCE, QUIVERS, AND VORTICES 7 their Hitchin Kobayashi correspondences (in the case of a holomorphic pair µ, consisting of a holomorphic vector bundle and a holomorphic section ¾ À ¼ µ, as considered in [B], which can be understood as a holomorphic triple Ç, the new parameter can actually be absorbed in, so no new parameters are really present). Second, the twisting bundles Å, for ¾ É ½, are not considered in [Ba, M]. Our method of proof combines the moment map techniques developed in [B, D2, S, UY] for bundles with a proof of a similar correspondence for quiver modules in [K, Ü6] Preliminaries and general notation. Throughout Section 3, Ê µ is a fixed holomorphic (Å-twisted) É-bundle with ʵ ¼. To prove Theorem 3.1, we can assume that É É ¼ É ½ µ is a finite quiver, with Ú ¼, for Ú ¾ É ¼, and ¼, for ¾ É ½ (if this is not the case, we remove the vertices Ú with Ú ¼, and the arrows with ¼, see Remark 1.2). The technical details of the proof largely simplify by introducing the following notation. Unless otherwise stated, Ú Ú ¼ (resp. ¼ ) stand for elements of É ¼ (resp. É ½ ), while sums, direct sums and products in Ú Ú ¼ (resp. ¼ ) are over elements of É ¼ (resp. É ½ ). Thus, the condition ʵ ¼ is equivalent to È Ú Ú Ú µ È Ú Ú Ö Ú µ. Let (3.3) Ú Ú a vector Ù in the fibre Ü over Ü ¾, is a collection vectors Ù Ú in the fibre ÚÜ over, for each Ú ¾ É ¼. Let Ú Å ¼ Ú µ Å ¼½ Ú µ be the -operator of the holomorphic vector bundle Ú, and let (3.4) Ú Ú be the induced -operator on. A hermitian metric À Ú on Ú defines a unique Chern connection ÀÚ compatible with the holomorphic structure Ú ; the corresponding covariant derivative is ÀÚ ÀÚ Ú, where ÀÚ Å ¼ Ú µ Å ½¼ Ú µ is its ½ ¼µ-part. Thus, given Ù ¾ Å µ, Ùµ ¾ Å ½ µ ÚÅ ½ Ú µ is the collection of Ú -valued ½µ-forms Ùµµ Ú Ú Ù Ú µ, for each Ú ¾ É ¼ Metrics and associated bundles. Let ÅØ Ú be the space of hermitian metrics on Ú. A her- mitian metric µ ÀÚ on Ú is determined by a smooth morphism À Ú Ú Ú, by Ù Ú Ù ¼ Ú µ À Ú À Ú Ù Ú µ Ù ¼ Ú µ, with Ù Ú Ù ¼ Ú in the same fibre of Ú. The right action of the complex gauge group Ú on ÅØ Ú is given, by means of this correspondence, by ÅØ Ú Ú ÅØ Ú, À Ú Ú µ À Ú Æ Ú. Let Ë Ú À Ú µ be the space of À Ú -selfadjoint smooth endomorphisms of Ú, for each À Ú ¾ ÅØ Ú. We choose a fixed hermitian metric Ã Ú ¾ ÅØ such that the hermitian metric Ø Ã Ú µ induced by Ã Ú on the determinant bundle Ø Ú µ satisfies ½ Ø ÃÚµ Ú µ, for each Ú ¾ É ¼ (such hermitian metric Ã Ú exists by Hodge theory). Any other metric on Ú is given by À Ú Ã Ú Ú for some Ú ¾ Ë Ú, or equivalently, by Ù Ú Ù ¼ Ú µ À Ú Ú Ù Ú Ù ¼ Ú µ à Ú, where Ë Ú Ë Ú Ã Ú µ. Let ÅØ be the space of hermitian metrics on such that the direct sum Ú Ú is orthogonal. A metric À ¾ ÅØ is given by a collection of metrics À Ú ¾ È ÅØ Ú, by Ù Ù ¼ µ À Ú Ù Ú Ù ¼ Ú µ À Ú. Let Ë Àµ ÚË Ú À Ú µ, for each À ¾ ÅØ, and Ë Ë Ãµ ÚË Ú. A vector ¾ Ë Àµ is given by a collection of vectors Ú ¾ Ë Ú À Ú µ, for each Ú ¾ É ¼, while a metric À ¾ ÅØ is given by À à for some ¾ Ë, i.e. À Ú Ã Ú Ú. The (fibrewise) norm on Ú (resp. ) corresponding to À Ú (resp. À), is given by Ù Ú ÀÚ Ù Ú Ù Ú µ ½¾ À Ú (resp. Ù À Ù Ùµ ½¾ À ). The corresponding ľ -metric and Ä ¾ -norm on the space of sections of Ú (resp. ), is defined by Ù Ú Ù ¼ Ú µ Ä ¾ À Ú (resp. Ù Ù ¼ µ Ä ¾ À È Ú Ù Ú Ù ¼ Ú µ Ä ¾ À Ú, Ù Ä ¾ À sections of, given by Ù Ú Ù ¼ Ú µ À Ú Ù Ú Ä ¾ À Ú Ù Ú Ù Ú µ ½¾ Ä ¾ À Ú ÓÖ Ù Ú Ù ¼ Ú ¾ Å ¼ Ú µ Ù Ä À Ù Ùµ½¾ Ä ¾ À ). The Ä -norm on the space of Ù À ½ ÓÖ Ù ¾ Å ¼ µ

9 8 LUIS ÁLVAREZ CÓNSUL AND OSCAR GARCÍA PRADA will also be useful. These metrics and norms induce canonical metrics on the associated bundles, which will be denoted with the same symbols. For instance, À Ú ¾ ÅØ Ú (resp. À ¾ ÅØ) induces an Ä -norm Ä À Ú on Ë Ú À Ú µ (resp. Ä À on Ë Àµ). To simplify the notation, we set Ù Ú Ù ¼ Ú µ Ù Ú Ù ¼ Ú µ Ã Ú Ù Ú Ù Ú ÃÚ Ù Ù ¼ µ Ù Ù ¼ µ à ٠٠à ; and Ù Ú Ù ¼ Ú µ Ä ¾ Ù Ú Ù ¼ Ú µ Ä ¾ Ã Ú Ù Ú Ä ¾ Ù Ú Ä ¾ Ã Ú Ù Ù ¼ µ Ä ¾ Ù Ù ¼ µ Ä ¾ Ã, Ù Ä Ù Ä Ã. The morphisms Ø Å Å induce a section of the representation bundle, defined as the smooth vector bundle over Ê Å ÀÓÑ Ø Å Å µ A metric À ¾ ÅØ induces another metric À on each term ÀÓÑ Ø Å Å µ of Ê, by ¼ µ À ØÖ Æ ¼ À µ for ¼ in the same fibre of ÀÓÑ Ø µ, where ¼ À Ø Å Å is defined as in È Ü2.1. Thus, À defines a hermitian metric on Ê, which we shall also denote À, by ¼ µ À ¼ µ À where ¼ are in a fibre of Ê. The corresponding fibrewise norm À is given by À µ ½¾. By integrating the hermitian metric over, À µ À and µ À induce Ä ¾ -inner products µ ÀÄ ¾ and µ ÀÄ ¾ on ż Ø ÅÅ µ and Å ¼ Å ¼ ʵ respectively, given by ¼ µ À Ä ¾ Ê µ À ÓÖ ¼ ¾ Å ¼ Ø Å Å µ and ¼ µ ÀÄ ¾ È µ Ä ¾ À ÓÖ ¼ ¾ Å ¼, with associated Ä ¾ -norms ÀÄ ¾, ÀÄ ¾ given by Ä ¾ À µ ½¾ Ä ¾ À and Ä ¾ À µ½¾ Ä ¾ À. We set ¼ µ ¼ µ Ã, Ã, for each ¼ in the same fibre of Ê; and ¼ µ Ä ¾ ¼ µ Ä ¾ Ã, Ä ¾ Ä ¾ Ã, for each ¼ smooth sections of Ê The vortex equations. Composition of two endomorphisms ¼ ¾ Ë is defined by Æ ¼ µ Ú Ú Æ ¼ Ú for Ú ¾ É ¼. The identity endomorphism of is given by Ú Ú. Given a vector bundle on, we define the endomorphisms Å Ò µ Å Ò µ, where Ò µ is the bundle of smooth endomorphisms of, by fibrewise multiplication, i.e. Å µµ Ú Å Ú Ú and Å µµ Ú Å Ú Ú, for ¾ and ¾ Ò µ in the fibres over the same point Ü ¾. For instance, if ¾ Ë, then µµ Ú Ú Ú Ú µ. Given À ¾ ÅØ and sections ¼ of Ê, we define the endomorphisms Æ ¼ À À Æ ¼ ¼ À ¾ Å ¼ Ò µµ, using Ü2.1, by Æ ¼ À µ Ú Ú¾ ½ µ Æ ¼ À À Æ ¼ µ Ú ¼ À Æ ¼ À À Æ ¼ Ú¾Ø ½ µ À Æ ¼ Note that À ¾ Ë Àµ. The quiver vortex equations (2.2) can now be written in a compact form (3.5) ½ À À ÓÖ À ¾ ÅØ Given ¾ Ë and ¾ Å ¼ Å ¼ ʵ, Æ Æ ¾ Å ¼ are defined by Æ µ Æ Æ µ Æ Ø Å Å µ Æ Æ Æ Æ The trace and trace free parts of the vortex equations. The trace map is defined by ØÖ Ò µ, ØÖ µ È Ú ØÖ Úµ. Let Ë ¼ Àµ be the space of -trace free À-selfadjoint endomorphisms ¾ Ë Àµ, i.e. such that ØÖ µ ¼, or more explicitly, È Ú Ú ØÖ Ú µ ¼, for each À ¾ ÅØ; let Ë ¼ Ë ¼ õ Ë. Let ÅØ ¼ be the space of metrics À à with ¾ Ë ¼. The metrics À ¾ ÅØ ¼ satisfy the trace part of equation (3.5), i.e. (3.6) ØÖ ½ À µ ØÖ µ To prove this, let À à ¾ ÅØ with ¾ Ë. Then Ø À Ú µ Ø Ã Ú µ ØÖ Ú so ØÖ ÀÚ Ø ÀÚµ Ø ÃÚµ ØÖ Ú ØÖ ÃÚ ØÖ Ú (since the operators induced by Ø Úµ and Ø ÃÚµ on the trivial bundle of endomorphisms of Ø Ú µ are and, resp.). Adding for all Ú,

10 HITCHIN KOBAYASHI CORRESPONDENCE, QUIVERS, AND VORTICES 9 ØÖ ½ À µ ØÖ ½ à µ ½ ØÖ µ, where ØÖ ½ ÃÚ µ Ú µ by construction (cf. Ü3.1.1), so ØÖ ½ à µ È Ú Ú Ú µ È Ú Ú Ö Ú µ ØÖ µ. Thus, (3.7) ØÖ ½ À µ ½ ØÖ µ which is zero if ¾ Ë ¼. This proves (3.6). Therefore, a metric À à ¾ ÅØ ¼ satisfies the quiver µ-vortex equations (3.5) if and only if it satisfies the trace free part, i.e. ¼ À ½ À À ¼ where ¼ À Ë Àµ Ë Àµ is the À-orthogonal projection onto ˼ Àµ Sobolev spaces. Following [UY, S, B], given a smooth vector bundle, and any integers ¼, Ä Å µ is the Sobolev space of sections of class Ä, i.e. -valued µ-forms whose derivatives of order have finite Ä -norm. Throughout the proof of Theorem 3.1, we fix an even integer ÑÊ µ ¾Ò. Note that there is a compact embedding of Ä ¾ Å µ into the space of continuous -valued µ-forms on, for ¾Ò. This embedding will be used in Ü Particularly important are the collection Ä ¾ Ë ÚÄ ¾ Ë Ú of Sobolev spaces Ä ¾ Ë Ú of Ã Ú -selfadjoint endomorphisms of Ú of class Ä ¾ ; the collection É ÅØ ¾ Ú ÅØ ¾Ú of Sobolev metrics, with ÅØ ¾Ú Ã Ú Ú Ú ¾ Ä ¾ Ë Ú ÓÖ Ú ¾ É ¼ the subspace Ä ¾ ˼ Ä ¾ Ë of sections ¾ Ä ¾Ë such that ØÖ µ ¼ almost everywhere in ; and ÅØ ¼ ¾ Ã Ú ¾ Ä ¾ ˼ ÅØ ¾ Given À à ¾ ÅØ ¾, with ¾ Ä ¾Ë, we define the À-adjoint of, generalizing the case where Ú is smooth, i.e. À Æ Ã Æ. Similar generalizations apply to the other constructions in ÜÜ3.1.2, 3.1.3, to define Ä ¾ Ë Ú À Ú µ and Ä ¾ Ë Àµ ÚÄ ¾ Ë Ú Àµ, as well as the subspace Ä ¾ ˼ Àµ Ä ¾ Ë Àµ, for each À ¾ ÅØ ¾. If À Ú Ã Ú Ú ¾ ÅØ ¾Ú with Ú ¾ Ä ¾ Ë Ú, we define the connection ÀÚ, with Ä ½ coefficients, and its curvature À Ú ¾ Ä Å ½½ Ò Ú µµ, with Ä coefficients, generalizing the case where Ú is smooth: (3.8) ÀÚ ÃÚ Ú ÃÚ Ú µ ÀÚ ÃÚ Ú Ú ÃÚ Ú µµ (where ÀÚ is the covariant derivative associated to the connection ÀÚ ) The degree of a saturated subsheaf. A saturated coherent subsheaf ¼ of a holomorphic vector bundle on (i.e., a coherent subsheaf with ¼ torsion-free), is reflexive, hence a vector subbundle outside of codimension 2. Given a hermitian metric À on, the À-orthogonal projection ¼ from onto ¼, defined outside codimension 2, is an Ä ¾ ½-section of the bundle of endomorphisms of, so ¼ µ is of class Ä ¾, where is the -operator of. The degree of ¼ is (cf. [UY, S, B]). ¼ ½ µ ÎÓÐ µ ØÖ ¼ ½ À µ ¾ Ä ¾ À Some constructions involving hermitian matrices. The following definitions slightly generalize [S, Ü4]. Let ³ Ê Ê and Ê Ê Ê be smooth functions. Given ¾ Ë, we define ³ µ ¾ Ë and linear maps µ Ë Ë and µ Å ¼ ʵ Å ¼ Ê) (we denote the last two maps with the same symbol since there will not be possible confusion between them). Actually, we define maps of fibre bundles Ë Ë Ò µ and Ë Ë Ò Êµ, for certain spaces Ë Ò µ and Ë Ò Êµ, which we first define. Let Ë Ò µ ÚË Ò Ú µ, where Ë Ò Ú µ is the space of smooth sections of the bundle Ò Ò Ú µ which are selfadjoint w.r.t. the metric induced by à Ú. Let Ò Ê be the endomorphism bundle of the vector bundle Ê; Ë Ò Êµ is the space of smooth sections of Ò Ê which are selfadjoint w.r.t. the metric induced by Ã Ú and Õ. We

11 10 LUIS ÁLVAREZ CÓNSUL AND OSCAR GARCÍA PRADA define ³ Ú µ ¾ Ë Ú for Ú ¾ Ë Ú and a linear map Ë Ú Ë Ò Ú µ as follows. Let Ú ¾ Ë Ú. If Ü ¾ È, let Ù Ú µ be an orthonormal basis of ÚÜ (w.r.t. Ã Ú ), with dual basis Ù Ú µ, such that Ú ÚÙ Ú Å Ù Ú. Furthermore, let Ñ µ be the dual of an orthonormal basis of Å Ü (w.r.t. Õ ). The value of ³ Ú µ ¾ Ë Ú at the point Ü ¾ is defined as in [S, Ü4], by (3.9) ³ Ú µ ܵ ³ Ú µù Ú Å Ù Ú We define ³ µ ¾ Ë, for ¾ Ë, by ³ µ Ú ³ Ú µ. Given Ú ¾ È Ë Ú with Ú Üµ ÚÙ Ú Å Ù Ú, the value of Ú µ Ú ¾ Ë Ú at the point Ü ¾ is (3.10) Ú µ Ú Üµ Ú Ú µ Ú Ù Ú Å Ù Ú and we define Ë Ë Ò µ and Ë Ë Ò Êµ as follows. Let ¾ Ë. First, if ¾ Ë, µ µ Ú Ú µ Ú. Second, given a section of Ê such that the value of Ø Å Å at Ü ¾ is ܵ È ÜµÙ ÅÙ Ø ÅÑ for each ¾ É ½, the value of µ ¾ Å ¼ ʵ at Ü ¾ is (3.11) µ ܵµ Ø µ ÜµÙ Å Ù Ø Å Ñ Note that if is given by Ü Ýµ ³ ½ ܵ³ ¾ ݵ for certain functions ³ ½ ³ ¾ µµ ³ ½ µ Æ Æ ³ ¾ µ Å Å µ, that is, (3.12) µ ³ ½ µ Æ Æ ³ ¾ µ Finally, given a smooth function ³ Ê Ê, we define ³ Ê Ê Ê as in [S, Ü4]: Thus, ³ Ü Ýµ ³ ݵ ³ ܵ Ü Ý Ò ³ Ü Ýµ ³ ¼ ܵ Ü Ý Ý Ü (3.13) ³ µµ ³ µ µµ ÓÖ ¾ Ë ÓÖ ¾ É ½ Ê Ê, then The following lemma will be especially important in the proof of Lemma Given a number, Ä ¾ Ë Ä Ë is the closed subset of sections ¾ ľ Ë such that a.e. in ; ľ ¼Ë Ò Êµ is similarly defined. Lemma (i) ³ Ë Ë extends to a continous map ³ Ä ¾ ¼ Ë Ä¾ ¼ ¼Ë for some ¼. (ii) ³ Ë Ë extends to a map ³ Ä ¾ ½ Ë ÄÕ ½ Ë for some ¼, for Õ ¾, which is continuous ¼ for Õ ¾. Formula (3.13) holds in this context. (iii) Ë Ë Ò µ extends to a map Ä ¾ ¼ Ë ÀÓÑ Ä¾ Å ¼ Ò µ Ä Õ Å ¼ Ò µµ for Õ ¾, which is continous in the norm operator topology for Õ ¾. (iv) Ë Ë Ò Êµ extends to a continuous map ³ Ä ¾ ¼ Ë Ä¾ ¼ ¼ Ë Ò Êµ for some ¼. (v) The previous maps extend to smooth maps ³ Ä ¾ Ë Ä ¾ Ë, Ä ¾ Ë Ä ¾Ë Ò µ and Ä ¾ Ë Ä ¾Ë Ò Êµ between Banach spaces of Sobolev sections. Formulas (3.9)-(3.13) hold everywhere in. Proof. This follows as in [B, S]. For (v), ¾Ò, so there is a compact embedding Ä ¾ ¼ Existence of special metric implies polystability. Let À be a hermitian metric on Ê satisfying the quiver µ-vortex equations. To prove that Ê is µ-polystable, we can assume that it is indecomposable then we have to prove that it is actually µ-stable. Let Ê ¼ ¼ ¼ µ Ê be a proper É-subsheaf. We can assume that ¼ Ú Ú is saturated for each Ú ¾ É ¼ (cf. Remark 2.6(iii)). Let ¼ Ú be the À Ú -orthogonal projection from Ú onto ¼ Ú, defined outside codimension 2,

12 HITCHIN KOBAYASHI CORRESPONDENCE, QUIVERS, AND VORTICES 11 Ú ¼¼ ¼ Ú, and Ú Ú ¼ µ. The collections of sections ¼ Ú Ú ¼¼ Ú define elements ¼ ¼¼ ¾ Ä ¾ ½ ż Ò µ, ¾ Ä ¾ Å ¼½ Ò µ, respectively. Taking the Ä ¾ -product with ¼ in (3.5), ½ À ¼ µ Ä ¾ À À ¼ µ Ä ¾ À ¼ µ Ä ¾ À We now evaluate the three terms of this equation. The first term in the left hand side is ½ À ¼ µ Ä ¾ À Ú ½ ÀÚ Ú ¼ µ Ä ¾ À Ú ÎÓÐ µ Ú Ú Ú Ú µ Ú Ú Ú ¾ Ä ¾ À Ú (cf. Ü3.1.5). Let ¼ ¼ ÆÆ ¼, ¼¼ ¼¼ ÆÆ ¼, ¼ ÆÆ ¼¼. Then ¼ Æ ¼ Æ ¼¼ ¼¼ Æ ¼¼ outside of codimension 2, for Ê ¼ Ê. Thus, ¼ Æ ¼¼, and the second term is À ¼ µ Ä ¾ À ¼ µ Ä ¾ À µ Ä ¾ À ¾ Ä ¾ À Finally, the right hand side is ¼ µ Ä ¾ À Ú Ú ØÖ Ú ¼ µ ÎÓÐ µ Ú Ö Ú ¼ µ (since ØÖ Ú ¼ µ Ö ¼ Ú µ outside of codimension 2). Therefore ÎÓÐ µ Ê ¼ µ Ú Ú ¾ Ä ¾ À Ú Ú ¾É ½ ¾ Ä ¾ À The indecomposability of Ê implies that either Ú ¼ for some Ú ¾ É ¼ or ¼ for some ¾ É ½ ; thus, Ê ¼ µ ¼, so Ê ¼ µ ¼ ʵ, hence Ê is µ-stable The modified Donaldson lagrangian. To define the modified Donaldson Lagrangian, we first recall the definition of the Donaldson lagrangian (cf. [S, Ü5]). Let Ê Ê Ê be given by (3.15) Ü Ýµ Ý Ü Ý Üµ ½ Ý Üµ ¾ The Donaldson lagrangian Å Ú Å Ã Ú µ ÅØ ¾Ú Ê is given by Å Ú À Ú µ ½ ÃÚ Ú µ Ä ¾ Ú µ Ú Ú µ Ú Ú µ Ä ¾ ÓÖ À Ú Ã Ú Ú ¾ ÅØ ¾Ú Ú ¾ Ä ¾ Ë Ú The Donaldson lagrangian Å Ú Å Ã Ú µ is additive in the sense that (3.16) Å Ú Ã Ú À Ú µ Å Ú À Ú Â Ú µ Å Ú Ã Ú Â Ú µ ÓÖ À Ú Â Ú ¾ ÅØ ¾ Another important property is that the Lie derivative of Å Ú at À Ú Ú ¾ Ä ¾ Ë Ú À Ú µ, is given by the moment map (cf. Ü2.2), i.e. (3.17) Å Ú À Ú Ú µ Higher order Lie derivatives can be easily evaluated. Thus, from (3.8), (3.18) ¾ ÅØ ¾, in the direction of ¼ ½ ÀÚ Ú µ Ä ¾ À Ú ÛØ À Ú ¾ ÅØ ¾ Ú ¾ Ä ¾ Ë Ú À Ú µ À Ú Ú Ú ÀÚ Ú Ú ÓÖ À Ú ¾ ÅØ ¾ Ò Ú ¾ Ä ¾ Ë À Úµ so the second order Lie derivative is ¾ (3.19) ¼ ½ Ú ÀÚ Ú Ú µ Ä ¾ À Ú Ú Ú Ä ¾ À Ú ¾ Å Ú À Ú Ú µ (the second equality is obtained by integrating ØÖ Ú ½ Ú ÀÚ Ú µ ½ ØÖ Ú ÀÚ Ú µ Ú Ú ¾ À Ú over,where Ú Ú ¾ À Ú ½ ØÖ Ú Ú ÀÚ Ú µ by the Kähler identities, and Ê ØÖ Ú ÀÚ Ú µ Ê ØÖ Ú ÀÚ Ú µµ Ò ½ Ò ½µ ¼ by Stokes theorem cf. e.g. [S, Lemma 3.1(b) and the proof Proposition 5.1]).

13 12 LUIS ÁLVAREZ CÓNSUL AND OSCAR GARCÍA PRADA Definition The modified Donaldson lagrangian Å Å Ã µ ÅØ ¾ Ê is Å Àµ Ú Ú Å Ú À Ú µ ¾ Ä ¾ À ¾ Ä ¾ à µ Ä ¾ ÓÖ À à ¾ ÅØ ¾ ¾ Ä ¾ Ë Using the constructions of Ü3.1.6, the modified Donaldson lagrangian can be expressed in terms of the functions Ê Ê Ê, with given by (3.15) and defined by (3.21) Ü Ýµ Ü Ý In the following, we use the notation µ Ä ¾ µ Ä ¾ Ã, Ä ¾ Ä ¾ Ã, as defined in Ü Lemma If À à ¾ ÅØ ¾, with ¾ Ä ¾Ë, then Å Àµ ½ à µ Ä ¾ µ µ µ Ä ¾ µ µ Ä ¾ ¾ Ä ¾ µ Ä ¾ Proof. The first two terms follow from the definitions of Å Ú and Å. To obtain the third term, we note that À Ø Å Å µ Æ Ã Æ and µµ Æ Æ Ø Å Å µ (cf. (3.12)), so ¾ À ØÖ Æ À µ ØÖ Æ Æ Ø Å Å µæ à µ ØÖ µµ Æ Ã µ µµ µ Ã. The last two terms follow directly from the definition of Å Minima of Å, the main estimate, and the vortex equations. Let Ñ ÅØ ¾ Ä Å ¼ Ò µ be defined by (3.23) Ñ Àµ ½ À À ÓÖ À à ¾ ÅØ ¾ ¾ Ä ¾ Ë Thus, Ñ Àµ ¾ Ä Ë Àµ for each À ¾ ÅØ ¾, and actually Ñ Àµ ¾ Ä Ë ¼ Àµ if À ¾ ÅØ ¼ ¾, by (3.6). Let Ñ Ãµ Ä be a positive real number. We are interested in the minima of Å in the closed subset of ÅØ ¼ ¾ ÅØ ¼ ¾ defined by À ¾ Åؼ Ñ ¾ Àµ Ä À (the restriction to this subset will be necessary to apply Lemma 3.33 below). Proposition If Ê is simple, i.e. its only endomorphisms are multiples of the identity, and À ¾ ÅØ ¼ ¾ minimises Å on ÅØ ¼ ¾, then Ñ Àµ ¼. The minima are thus the solutions of the vortex equations. To prove this, we need a lemma about the first and second order Lie derivaties of Å. Given À ¾ ÅØ ¾, Ä À Ä ¾ Ë Àµ Ä Ë Àµ is defined by (3.25) Ä À µ Since À À, with À À, we have Ñ À µ ¼ ÓÖ ¾ Ä ¾ Ë Àµ (3.26) À ¼ À so À ¼ À. Together with (3.18), this implies that (3.27) Ä À µ ½ À À Lemma (i) Å Ã Àµ Å À µ Šà µ, for À  ¾ ÅØ ¾ ; (ii) (iii) Å À µ ¼ Ñ Àµ µ Ä ¾ À, for each À ¾ ÅØ ¾ and ¾ Ä ¾ Ë Àµ; ¾ Å ¾ À µ ¼ Ä À µ µ Ä ¾ À ÅØ ¾ and ¾ Ä ¾ Ë Àµ. Ú Ú Ú Ú ¾ Ä ¾ À Ú ¾ Ä ¾ À, for each À ¾

14 HITCHIN KOBAYASHI CORRESPONDENCE, QUIVERS, AND VORTICES 13 Proof. Part (i) follows immediately from (3.16) and à µ ¼ à ¼. To prove (ii) and (iii), let À À, for ¾ Ê. From (3.26) we get ¾ À ¼ ØÖ À ¼ ØÖ À µ À µ À, which together with (3.17), proves (ii) (the last term in (3.23) is trivially obtained). The first equality in (iii) follows from (ii), the À -selfadjointness of (since À À ), and (3.25): ¾ Å ¾ À µ ¼ Ñ À µ µ Ä ¾ À ¼ ØÖ Ñ À µ ¼ ØÖ Ä À µ µ which equals Ä À µ µ Ä ¾ À. To prove the second equality in (iii), we first notice that if ¼ is a smooth section of Ê, then ¼ Æ À µ À Æ ¼ µ À and À Æ ¼ µ À Æ ¼ µ À, so ¼ À µ À ¼ µ À. The second equality in (iii) is now obtained using (3.27), (3.18) and taking ¼ in the previous formula. Proof of Proposition We start proving that if Ê is simple and À ¾ ÅØ ¼ ¾, then the restriction of Ä À to Ä ¾ ˼ Àµ, which we also denote by Ä À Ä ¾ ˼ Àµ Ä Ë ¼ Àµ, is surjective. To do this, we only have to show that Ä À is a Fredholm operator of index zero and that it has no kernel. First, for each vertex Ú, Ú Ä ¾ Ë Ú À Ú µ Ä Ë Ú À Ú µ, defined by Ú ½ Ú ÀÚ ½ Ú ÃÚ, is obviously a compact operator (cf. Ü3.1.4), and by the Kähler identities, ½ Ú ÃÚ acting on Ä ¾ Ë is the ½ ¼µ-laplacian ¼ Ã Ú ÃÚ Ã Ú È Ã Ú ÃÚ, which is elliptic and selfadjoint, hence Fredholm, and has index zero. Now, Ä À equals Ú Ú ½ ÀÚ, up to a compact operator, so it is also a Fredholm operator of index zero. To prove that it has no kernel, we notice that if ¾ Ä ¾ ˼ Àµ satisfies Ä À µ ¼, then Ä À µµ Ä ¾ À ¼, so Lemma 3.28(iii) implies Ú Ú ¼ and ¼; i.e. is actually an endomorphism of Ê, so Ú Ú, for certain constant. Since ØÖ µ ¼, the constant is ¼, so Ú ¼. Let À minimise Å in ÅØ ¼ ¾. To prove that Ñ Àµ ¼, we assume the contrary. Since Ä À Ä ¾ ˼ Àµ Ä Ë ¼ Àµ is surjective, and Ñ Àµ ¾ Ë ¼ Àµ is not zero, there exists a nonzero ¾ Ä ¾ ˼ Àµ with Ä À µ Ñ Àµ. We shall consider the values of Å along the path À À ¾ ÅØ ¼ ¾ for small. First, Ñ À µ ¾ ¼ À ØÖ Ñ À µ ¾ µ (cf. (3.25)), and since is even, Ñ À µ Ä À ¼ ¾ Ñ Àµ ¾ À ¼ ¾ Ñ Àµ Ä À µµ À ¾Ñ Àµ ¾ À Ñ À µ ¾ À ¼ Ñ Àµ Ä À ¼ so the path À is in ÅØ ¼ ¾ for small. Thus, Å À µ ¼ ¼, as À minimises Å in ÅØ ¼ ¾. Now, Lemma 3.28(ii) applied to ¾ Ä ¾Ë Àµ gives Å À µ ¼ Ñ Àµ µ Ä ¾ À Ä À µ µ Ä ¾ À As in the first paragraph of this proof, if Ê is simple and ¾ Ä ¾ ˼ Àµ satisfies Ä À µµ Ä ¾ À ¼, then Lemma 3.28(iii) implies that is zero. This contradicts the assumption Ñ Àµ ¼. Definition We say that Å satisfies the main estimate in ÅØ ¼ ¾ if there are constants ½ ¾ ¼, which only depend on, such that Ù ½ Å Àµ ¾, for all À à ¾ ÅØ ¼ ¾, ¾ Ä ¾ Ë. Proposition If Ê is simple and Å satisfies the main estimate in ÅØ ¼ ¾, then there is a hermitian metric on Ê satisfying the µ-vortex equations. This hermitian metric is unique up to multiplication by a positive constant.

15 14 LUIS ÁLVAREZ CÓNSUL AND OSCAR GARCÍA PRADA Proof. This result is proved in exactly the same way as in [B, Ü3.14], so here we only sketch the proof. One first shows that if Å Ã µ is bounded above, then the Sobolev norms Ä are ¾ bounded. One then takes a minimising sequence à for Å, with ¾ Ä ¾ ˼ ; then Ä ¾ are uniformly bounded, so after passing to a subsequence, converges weakly in Ä ¾ to some. One then sees that Å is continuous in the weak topology on ÅØ ¼ ¾, so Å Ã µ converges to Å Ã µ. Thus, À à minimises Å. By Proposition 3.24, Ñ Àµ ¼, i.e. À satisfies the vortex equations. By elliptic regularity, À is smooth. The uniqueness of the solution À follows from the convexity of Å (cf. Lemma 3.28(iii)) and the simplicity of Ê. The proof of Theorem 3.1 is therefore reduced to show that if Ê is µ-stable, then Å satisfies the main estimate in ÅØ ¼ ¾ (this is the content of Ü3.6) Equivalence of ¼ and Ä ½ estimates. The following proposition will be used in Ü3.6. Proposition There are two constants ½ ¾ ¼, depending on and, such that for all À à ¾ ÅØ ¼ ¾, ¾ Ä ¾ ˼, Ù ½ Ä ½ ¾. Corollary Å satisfies the main estimate in ÅØ ¼ ¾ if and only if there are constants ½ ¾ ¼, which only depend on, such that Ä ½ ½ Å Àµ ¾, for all À à ¾ ÅØ ¼ ¾, ¾ Ä ¾ ˼. Corollary 3.32 is immediate from Proposition To prove Proposition 3.31, we need three lemmas. The first one is due to Donaldson [D3] (see also the proof of [S, Proposition 2.1]). Lemma There exists a smooth function ¼ ½µ ¼ ½µ, with ¼µ ¼ and ܵ Ü for Ü ½, such that the following is true: For any ¾ Ê, there is a constant µ such that if is a positive bounded function on and, where is a function in Ä µ Òµ with Ä, then Ù µ Ä ½µ. Furthermore, if ¼, then ¼. Lemma If ¾ Ä ¾ Ë and À à ¾ ÅØ ¾, then À µ à µ. Proof. The function µ À µ for ¾ Ê, where À Ã, is increasing, as µ ¾ À ¼ (cf. (3.26)). Now, ¼µ à µ, ½µ À µ, so we are done. Lemma If À à ¾ ÅØ ¾, with ¾ Ä ¾Ë, then Ñ Àµ Ñ Ãµ µ ½ ¾ ½¾ ½¾ where ½¾ ¾ Ä ¾ Ë is of course defined by ½¾ µ Ú ½¾ Ú Ú, for Ú ¾ É ¼. Proof. This lemma, and its proof, are similar to (but not completely immediate from) [B, Proposition 3.7.1]. First, Lemma 3.34 and (3.8) imply (3.36) Ñ Àµ Ñ Ãµ µ ½ À à µ ½ à µ µ where (3.37) à µ µ à µ à à µ (for à is the Chern connection corresponding to the metric Ã). To make some local calculations, we choose a local Ã Ú -orthogonal basis Ù Ú of eigenvectors of Ú, for each vertex Ú, with corresponding eigenvalues Ú, and let Ù Ú be the corresponding dual basis; thus, Ú Ú Ù Ú Å Ù Ú

16 HITCHIN KOBAYASHI CORRESPONDENCE, QUIVERS, AND VORTICES 15 As in [B, (3.36)], a local calculation gives Ú ÃÚ Ú Ú µ ½ ¾ Ú ¾ ; multiplying by Ú and adding for Ú ¾ É ¼, we get à µ ½ ¾ ¼ ¾, where ¼ ½¾. Thus, (3.38) à µ ½ ¾ ¼ ¾ ¼ ¼ ¼ ¼ From (3.36), (3.37), (3.38) and the equality ¾ ½ for the action of the laplacian on ¼-forms in a Kähler manifold, we get Ñ Àµ Ñ Ãµ µ ½ ¾ ¼ ¼ ½ ¼ µ ½ à à µ In the proof of [B, Proposition 3.7.1], there are several local calculations which, although there they are only used for the section ¾ Ä ¾ Ë defining the metric À Ã, are actually valid for any Ã-selfadjoint section, in particular for ¼ ¾ Ä ¾ Ë. Thus, [B, (3.42)] applied to Ú is ½ Ú ÃÚ Ú ÃÚ Ú µ ½ Ú Ú µ and multiplying by Ú and adding for Ú ¾ É ¼, we get (3.39) ½ à à µ Ú ½ ¼ Ú ¼ Ú µ where ¼ Ú (3.40) ½¾ Ú Ú Ú are the eigenvalues of ¼ Ú ½¾ Ú ; similarly, [B, (3.43)] applied to ¼ is Ú ½ ¼ Ú ¼ Ú µ ½ ¼ ¼ µ ½ ¼ ¼ µ From (3.38), (3.39), (3.40), we obtain Ñ Àµ Ñ Ãµ µ ½ ¾ ¼ ¼. Proof of Proposition Let ÑÒ ÑÒ Ú Ú ¾ É ¼, ÑÜ ÑÜ Ú Ú ¾ É ¼. Given À à ¾ ÅØ ¼ ¾, with ¾ Ä ¾ ˼, let ½¾ and ¾ ½¾ ÑÒ Ñ Àµ Ñ Ãµµ. We now verify that and satisfy the hypotheses of Lemma 3.33, for a certain which only depends on. First, Ä ¾ ½¾ ÑÒ Ñ Àµ Ä Ñ Ãµ Ä µ ½¾ ¾ ÑÒ ¾½. Second, we prove that (3.41) At the points where does not vanish, ½ ½¾ ½ ÑÒ, so Lemma 3.35 gives ¾ ½¾ ÑÒ ½ Ñ Àµ Ñ Ãµ µ ¾ ½¾ Ñ Àµ Ñ Ãµ while to consider the points where vanishes, we just take into account that ¼ almost everywhere (a.e.) in ½ ¼µ, and that ¼ by its definition, so (3.41) actually holds a.e. in. The hypotheses of Lemma 3.33 are thus satisfied, so there exists a constant µ ¼ such that Ù µ Ä ½µ, with with ¼ ½µ ¼ ½µ as in Lemma This estimate can also be written as Ù ½ Ä ½ ¾, where ½ ¾ ¼ only depend on. Now, ½¾ ÑÒ and ÑÜ, ½¾ so Ù ½¾ ÑÒ ½ Ä ½ ¾ µ ½¾ ÑÒ ÑÒ ½ÑÜ ½¾ Ä ½ ¾ µ The estimate is obtained by redefining the constants ½ ¾.

17 16 LUIS ÁLVAREZ CÓNSUL AND OSCAR GARCÍA PRADA 3.6. Stability implies the main estimate. The following proposition, together with Proposition 3.30, are the key ingredients to complete the proof of Theorem 3.1 (cf. Definition 3.29 for the main estimate). Proposition If Ê is µ-stable, then Å satisfies the main estimate in ÅØ ¼ ¾. To prove this, we need some preliminaries (Lemmas ). Let ½ ½ be a sequence of constants with ÐÑ ½. ½ Lemma If Å does not satisfy the main estimate in ÅØ ¼ ¾, then there is a sequence ½ ½ in Ä ¾ ˼ with à ¾ ÅØ ¼ ¾ (which we can assume to be smooth), such that (i) ÐÑ ½ Ä ½ ½, (ii) Ä ½ Å Ã µ. Proof. Let Ñ Ãµ Ä with, so ÅØ ¼ ¾ Åؼ ¾. Thus, if Å does not satisfy either. We shall the main estimate in ÅØ ¼ ¾, then it does not satisfy the main estimate in Åؼ ¾ prove that for any positive constant ¼, if there are positive constants ¼¼ and Æ such that Ä ½ ¼ Å Ã µ ¼¼ whenever ¾ Ä ¾ ˼ with à ¾ ÅØ ¼ ¾ and Ä ½ Æ, then Å satisfies the main estimate in ÅØ ¼ ¾. The lemma follows from this claim by choosing a sequence of constants Æ ½ ½ with Æ ½, and taking ¼¼ and ¾ Ä ¾ ˼ with à Æ, and Ä ½ Å Ã µ ¼¼. Let ¼ ¼¼ Æ be such that Ä ½ ¼ Å Ã µ ¼¼ for Ä ½ Æ ¾ ÅØ ¼ ¾ Åؼ ¾, Ä ½ Let Ë Æ ¾ Ä ¾ ˼ à ¾ ÅØ ¼ ¾ and Ä ½ Æ. By Proposition 3.31, if ¾ Ë Æ, then Ù Ú Ù ½ Ä ½ ¾ ½ Æ ¾ (here ½ and ¾ are not the first elements of the sequence ½ ½ but constants as in Proposition 3.31), so by Lemma 3.22, Å is bounded below on Ë Æ, i.e. Å Ã µ for each ¾ Ë Æ, for some constant ¼. Thus, Ä ½ ¼ Å Ã µ µ Æ for each ¾ Ë Æ. Replacing ¼¼ by ÑÜ ¼¼ ¼ Æ, we see that Ä ½ ¼ Å Ã µ ¼¼, for each ¾ Ä ¾ ˼ with à ¾ ÅØ ¼ ¾. By Corollary 3.32, Å satisfies the main estimate in ÅØ ¼ ¾. Finally, since the set of smooth sections is dense in Ä ¾ ˼, we can always assume that is smooth (we made the choice so that if à is in the boundary Ñ Àµ Ä À of Åؼ ¾, we can still replace by a smooth ¼ with à ¼ ¾ ÅØ ¼ ¾ ). Lemma Assume that Å does not satisfy the main estimate in ÅØ ¼ ¾. Let ½ ½ be a sequence as in Lemma 3.43, Ð Ä ½, µ ½ ¾, where ½ ¾ are as in Proposition 3.31, and Ù Ð. Thus, Ù Ä ½ ½ and Ù Ù µ. After going to a subsequence, Ù Ù ½ weakly in Ä ¾ ½ ˼, for some nontrivial Ù ½ ¾ Ä ¾ ˼ such that if Ê Ê Ê is a smooth non-negative function such that Ü Ýµ ½ Ü Ýµ whenever Ü Ý, and Ê Ê Ê is a smooth non-negative function with Ü Ýµ ¼ whenever Ü Ý, for some fixed ¼, then ½ à ٠½ µ Ä ¾ Ù ½ µ Ù ½ Ù ½ µ Ä ¾ µ µ Ä ¾ Ù ½ µ Ä ¾ ¼ Proof. To prove this inequality, we can assume that and have compact support (for Ù Ù are bounded, by Lemma 3.31, and the definitions of µ Ù ½ and µ only depend on the values of and at the pairs µ of eigenvalues, as seen in Ü3.1.6). Now, if and have compact support then, for large enough Ð, Ü Ýµ Ð ÐÜ Ðݵ Ü Ýµ Ð ½ ÐÜ Ðݵ

18 HITCHIN KOBAYASHI CORRESPONDENCE, QUIVERS, AND VORTICES 17 where and are defined as in (3.15) and (3.21) (cf. the proof of [B, Proposition 3.9.1]). Since Ð ½, from these inequalities we obtain that for large enough, Ù Ú µ Ù Ú Ù Ú µ Ä ¾ Ð Ð Ú Ù Ú µ Ù Ú Ù Ú µ Ä ¾ Ù µ µ Ä ¾ Ð ½ Ð Ù µ µ Ä ¾ so Lemma 3.43(iii) applied to Ð Ù, together with lemma 3.22, give an upper bound ½ ¾ Ä ¾ Ð Ð ½ Å Ã Ð Ù µ Ð ½ ¾ Ä ¾ ½ à ٠µ Ä ¾ Ù µ Ù Ù µ Ä ¾ Ù µ µ Ä ¾ Ù µ Ä ¾ As in the proof of [B, Proposition 3.9.1], one can use this upper bound to show that the sequence Ù ½ ½ is bounded in ľ ½. Thus, after going to a subsequence, Ù Ù ½ in Ä ¾ ½, for some Ù ½ ¾ Ä ¾ ½ Ë with Ù ½ Ä ½ ½, so Ù ½ is non-trivial. We now prove the estimate for Ù ½. First, since Ù Ù µ, Ù Ù ½ in Ä ¾ ¼ ; applying Lemma 3.14(iii), one can show (as in the proof of [S, Lemma 5.4]) that ½ à ٠µ Ä ¾ Ù µ Ù Ù µ Ä ¾ approaches ½ à ٠½ µ Ä ¾ Ù ½ µ Ù ½ Ù ½ µ Ä ¾ as ½. Second, since Ä ¾ ½ Ä ¾ is a compact embedding and actuallly Ù ¾ Ä ¾ ½ Ë Ä¾ ¼Ë, applying Lemma 3.14(iv) (as in the proof of [B, Proposition 3.9.1]), Ä ¾ ¼ Ë Ä¾ ¼ ¼ Ë Ò Êµ, Ù Ùµ, is continuous on Ä ¾ ¼ Ë, so ÐÑ ½ Ù µ Ù ½ µ. Since Ù Ù are bounded, this implies that Ù µ µ Ä ¾ converges to Ù ½ µ µ Ä ¾ as ½. Finally, it is clear that Ù µ Ä ¾ Ù ½ µ Ä ¾ as ½. This completes the proof. Lemma If Å does not satisfy the main estimate in ÅØ ¼ ¾, and Ù ½ ¾ Ä ¾ ˼ is an in Lemma 3.44, then the following happens: (i) The eigenvalues of Ù ½ are constant almost everywhere. (ii) Let the eigenvalues of Ù ½ be ½ Ö. If whenever, ½ Ö, then Ù ½ µ Ù ½ µ ¼. (iii) If is an in Proposition 3.44, then Ù ½ µ ¼. Ê Ê Ê satisfies µ ¼ Proof. Parts (i) and (ii) of are proved as in [UY, appendix], [S, ÜÜ6.3.4 and 6.3.5], or [B, ÜÜ3.9.2 and 3.9.3], using Lemma3.14(ii) for part (i) and the estimate in Lemma 3.44 for part (ii). Part (iii) is similar to [B, Lemma 3.9.4], and again uses the estimate in Lemma We now construct a filtration of quiver subsheaves of Ê using Ä ¾-subsystems, as in [B, Ü3.10]. Lemma Assume that Å does not satisfy the main estimate in ÅØ ¼ ¾. Let Ù ½ ¾ Ä ¾ ˼ be as in Lemma Let the eigenvalues of Ù ½, listed in ascending order, be ¼ ½ Ö. Since Ù ½ is -trace free (cf. Ü3.1.3), there are at least two different eigenvalues, i.e. Ö ½. Let ¼ Ö Ê Ê be smooth functions such that, for Ö, ܵ ½ if Ü, ܵ ¼ if Ü ½, and Ö Üµ ½ if Ü Ö. Let Ú Ú be the canonical projections (cf. (3.3)) and be as in (3.4). The operators ¼ Ö Ù ½ µ and ¼ Ú ¼ Æ Ú, for ¼ Ö, satisfy: (i) ¼ ¾ Ä ¾ ½ Ë, ¼¾ ¼ ¼ à and ½ ¼ µ ¼ ¼. (ii) ¼ µ Æ Æ ¼ Ø Å Å µ ¼ for each Ú ¾ É ¼. (iii) Not all the eigenvalues of Ù ½ are positive. Proof. The proof of (i) is as in [S] (right below Lemma 5.6; see also [B, Proposition (i)-(iii)]). Part (ii) is similar to, but more involved than, [B, Proposition (iv)], so we now give a detailed proof of this part. For each, let ¼ be such that ½ µ¾, and ³ ½ ³ ¾ Ê Ê be smooth non-negative functions such that ³ ½ ܵ ¼ if Ü ½ ¾ and ³ ½ ܵ ½ if Ü ½,