Oscar García-Prada

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant

Moduli spaces of holomorphic triples over compact Riemann surfaces

Abstract.A holomorphic triple over a compact Riemann surface consists of two holomorphic vector bundles and a holomorphic map between them. After fixing the topological types of the bundles and a real parameter, there exist moduli spaces of stable holomorphic triples. In this paper we study non-emptiness, irreducibility, smoothness, and birational descriptions of these moduli spaces for a certain range of the parameter. Our results have important applications to the study of the moduli space of representations of the fundamental group of the surface into unitary Lie groups of indefinite signature ([5, 7]). Another application, that we study in this paper, is to the existence of stable bundles on the product of the surface by the complex projective line.

Hitchin-Kobayashi correspondence, quivers, and vortices

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.

Betti numbers of the moduli space of rank 3 parabolic Higgs bundles

Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles with fixed and non-fixed determinant, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we carry out a careful analysis of them by studying their variation with this parameter. Thus we obtain in particular information about the topology of the moduli spaces of parabolic triples for the value of the parameter relevant to the study of parabolic Higgs bundles. The remaining critical submanifolds are also described: one of them is the moduli space of parabolic bundles, while the remaining ones have a description in terms of symmetric products of the Riemann surface. As another consequence of our Morse theoretic analysis, we obtain a proof of the parabolic version of a theorem of Laumon, which states that the nilpotent cone (the preimage of zero under the Hitchin map) is a Lagrangian subvariety of the moduli space of parabolic Higgs bundles.

Coherent systems and Brill-Noether theory

Let X be a curve of genus g. A coherent system on X consists of a pair (E,V), where E is an algebraic vector bundle over X of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the variation of the moduli space of coherent systems when we move the parameter. As an application, we analyze the cases k=1,2,3 and n=2 explicitly. For small values of α, the moduli spaces of coherent systems are related to the Brill–Noether loci, the subschemes of the moduli spaces of stable bundles consisting of those bundles with at least a prescribed number of independent sections. The study of coherent systems is applied to find the dimension, prove the irreducibility, and in some cases calculate the Picard groups of the Brill–Noether loci with k≤3.

Higgs bundles and surface group representations in the real symplectic group

In this paper, we study the moduli space of representations of a surface group (that is, the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n, R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor–Wood-type inequality. Our main result is a count of the number of connected components of the moduli space of maximal representations, that is, representations with maximal Toledo invariant. Our approach uses the non-abelian Hodge theory correspondence proved in a companion paper (O. Garc´oa-Prada, P. B. Gothen and I. Mundet i Riera, The Hitchin–Kobayashi correspondence, Higgs pairs and surface group representations, Preprint, 2012, arXiv:0909.4487 [math.AG].) to identify the space of representations with the moduli space of polystable Sp(2n, R)-Higgs bundles. A key step is provided by the discovery of new discrete invariants of maximal representations. These new invariants arise from an identification, in the maximal case, of the moduli space of Sp(2n, R)-Higgs bundles with a moduli space of twisted Higgs bundles for the group GL(n, R).

DIMENSIONAL REDUCTION,

In this paper we study gauge theory on

{\rm SL} (2, {\mathbb C})

{\rm SL} (2, {\mathbb C})

-EQUIVARIANT BUNDLES AND STABLE HOLOMORPHIC CHAINS

-equivariant bundles over X × ℙ1, where X is a compact Kahler manifold, ℙ1 is the complex projective line, and the action of

Homotopy groups of moduli spaces of representations

{\rm SL} (2, {\mathbb C})