The so-called Hitchin-Kobayashi correspondence, proved by Donaldson, Uhlenbeck and Yau, establishes that an indecomposable holomorphic vector bundle over a compact Kähler manifold admits a Hermitian-Einstein metric if and only if the bundle satisfies the Mumford-Takemoto stability condition. In this paper we consider a variant of this correspondence for G-equivariant vector bundles on the product of a compact Kähler manifold X by a flag manifold G/P , where G is a complex semisimple Lie group and P is a parabolic subgroup. The modification that we consider is determined by a filtration of the vector bundle which is naturally defined by the equivariance of the bundle. The study of invariant solutions to the modified Hermitian-Einstein equation over X × G/P leads, via dimensional reduction techniques, to gauge-theoretic equations on X. These are equations for hermitian metrics on a set of holomorphic bundles on X linked by morphisms, defining what we call a quiver bundle for a quiver with relations whose structure is entirely determined by the parabolic subgroup P . Similarly, the corresponding stability condition for the invariant filtration over X × G/P gives rise to a stability condition for the quiver bundle on X, and hence to a Hitchin-Kobayashi correspondence. In the simplest case, when the flag manifold is the complex projective line, one recovers the theory of vortices, stable triples and stable chains, as studied by Bradlow, the authors, and others.

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.

In this paper we study gauge theory on SL(2, C)-equivariant bundles over X × P 1 , where X is a compact Kähler manifold, P 1 is the complex projective line, and the action of SL(2, C) is trivial on X and standard on P 1 . We first classify these bundles, showing that they are in correspondence with objects on X -that we call holomorphic chainsconsisting of a finite number of holomorphic bundles E i and morphisms E i → E i−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 × P 1 to X.

We show how natural functors from the category of coherent sheaves on a projective scheme to categories of Kronecker modules can be used to construct moduli spaces of semistable sheaves. This construction simplifies or clarifies technical aspects of existing constructions and yields new simpler definitions of theta functions, about which more complete results can be proved.Comment: 52 pp. Dedicated to the memory of Joseph Le Potier. To appear in Inventiones Mathematicae. Slight change in the definition of the Kronecker algebra in Secs 1 (p3) and 2.2 (p6), with corresponding small alterations elsewhere, to make the constructions work for non-reduced schemes. Section 6.5 rewritten. Remark 2.6 and new references adde