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Oscar Garcia-Prada

Publications and source records attributed to Oscar Garcia-Prada.

At least 37 records · Page 2Linked to original sources
arXiv2013

Higgs bundles over elliptic curves

In this paper we study $G$-Higgs bundles over an elliptic curve when the structure group $G$ is a classical complex reductive Lie group. Modifying the notion of family, we define a new moduli problem for the classification of semistable $G$-Higgs bundles of a given topological type over an elliptic curve and we give an explicit description of the associated moduli space as a finite quotient of a product of copies of the cotangent bundle of the elliptic curve. We construct a bijective morphism from this new moduli space to the usual moduli space of semistable $G$-Higgs bundles, proving that the former is the normalization of the latter. We also obtain an explicit description of the Hitchin fibration for our (new) moduli space of $G$-Higgs bundles and we study the generic and non-generic fibres.

math.AG↗
arXiv2012

Pseudo-real principal Higgs bundles on compact Kaehler manifolds

Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form $σ_G$. We define pseudo-real principal $G$--bundles on $X$; these are generalizations of real algebraic principal $G$--bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal $G$--bundles. Their relationships with the usual stable, semistable and polystable principal $G$--bundles are investigated. We then prove that the following Donaldson--Uhlenbeck--Yau type correspondence holds: a pseudo-real principal $G$--bundle admits a compatible Einstein-Hermitian connection if and only if it is polystable. A bijection between the following two sets is established: 1) The isomorphism classes of polystable pseudo-real principal $G$--bundles such that all the rational characteristic classes of the underlying topological principal $G$--bundle vanish. 2) The equivalence classes of twisted representations of the extended fundamental group of $X$ in a $σ_G$--invariant maximal compact subgroup of $G$. (The twisted representations are defined using the central element in the definition of a pseudo-real principal $G$--bundle.) All these results are also generalized to the pseudo-real Higgs $G$--bundle.

math.AG↗
arXiv2012

The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations

We develop a complete Hitchin-Kobayashi correspondence for twisted pairs on a compact Riemann surface X. The main novelty lies in a careful study of the the notion of polystability for pairs, required for having a bijective correspondence between solutions to the Hermite-Einstein equations, on one hand, and polystable pairs, on the other. Our results allow us to establish rigorously the homemomorphism between the moduli space of polystable G-Higgs bundles on X and the character variety for representations of the fundamental group of X in G. We also study in detail several interesting examples of the correspondence for particular groups and show how to significantly simplify the general stability condition in these cases.

math.DG↗
arXiv2012

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 (i.e., 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, i.e. representations with maximal Toledo invariant. Our approach uses the non-abelian Hodge theory correspondence proved in a companion paper arXiv:0909.4487 [math.DG] 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).

math.AG↗
arXiv2010

A Milnor-Wood inequality for complex hyperbolic lattices in quaternionic space

We prove a Milnor-Wood inequality for representations of the fundamental group of a compact complex hyperbolic manifold in the group of isometries of quaternionic hyperbolic space. Of special interest is the case of equality, and its application to rigidity. We show that equality can only be achieved for totally geodesic representations, thereby establishing a global rigidity theorem for totally geodesic representations.

math.DG↗
arXiv2010

Deformations of maximal representations in Sp(4,R)

We use Higgs bundles to answer the following question: When can a maximal Sp(4,R)-representation of a surface group be deformed to a representation which factors through a proper reductive subgroup of Sp(4,R)?

math.AG↗
arXiv2010

Higgs bundles for the Lorentz group

Using the Morse-theoretic methods introduced by Hitchin, we prove that the moduli space of $\SO_0(1,n)$-Higgs bundles when $n$ is odd has two connected components.

math.AG↗
arXiv2008

Moduli spaces of parabolic $U(p,q)$-Higgs bundles

Using the $L^2$-norm of the Higgs field as a Morse function, we count the number of connected components of the moduli space of parabolic $U(p,q)$-Higgs bundles over a Riemann surface with a finite number of marked points, under certain genericity conditions on the parabolic structure. This space is homeomorphic to the moduli space of representations of the fundamental group of the punctured surface in $U(p,q)$, with fixed compact holonomy classes around the marked points. We apply our results to the study of representations of the fundamental group of elliptic surfaces of general type.

math.AG↗
arXiv2007

Homotopy groups of moduli spaces of representations

We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups GL(n,C) and U(p,q). Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott--Morse theory on the corresponding moduli spaces.

math.AG↗
arXiv2006

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.

math.AG↗
arXiv2005

On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces

We study holomorphic $(n+1)$-chains $E_n\to E_{n-1} \to >... \to E_0$ consisting of holomorphic vector bundles over a compact Riemann surface and homomorphisms between them. A notion of stability depending on $n$ real parameters was introduced in the work of the first two authors and moduli spaces were constructed by the third one. In this paper we study the variation of the moduli spaces with respect to the stability parameters. In particular we characterize a parameter region where the moduli spaces are birationally equivalent. A detailed study is given for the case of 3-chains, generalizing that of 2-chains (triples) in the work of Bradlow, Garcia-Prada and Gothen. Our work is motivated by the study of the topology of moduli spaces of Higgs bundles and their relation to representations of the fundamental group of the surface.

math.AG↗
arXiv2003

Moduli spaces of holomorphic triples over compact Riemann surfaces

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, which we explore in a companion paper "Surface group representations in PU(p,q) and Higgs bundles". 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. This paper, and its companion mentioned above, form a substantially revised version of math.AG/0206012.

math.AG↗
arXiv2003

Surface group representations in PU(p,q) and Higgs bundles

Using the L^2 norm of the Higgs field as a Morse function, we study the moduli spaces of U(p,q)-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on (p,q). A key step is the identification of the function's local minima as moduli spaces of holomorphic triples. In a companion paper "Moduli spaces of holomorphic triples over compact Riemann surfaces" (math.AG/0211428) we prove that these moduli spaces of triples are non-empty and irreducible. Because of the relation between flat bundles and fundamental group representations, we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple PU(p,q)-representations. The topological invariants of the flat bundles are used to label subspaces. These invariants are bounded by a Milnor-Wood type inequality. For each allowed value of the invariants satisfying a certain coprimality condition, we prove that the corresponding subspace is non-empty and connected. If the coprimality condition does not hold, our results apply to the closure of the moduli space of irreducible representations. This paper, and its companion mentioned above, form a substantially revised version of math.AG/0206012.

math.AG↗
arXiv2002

Relative Hitchin--Kobayashi correspondences for principal pairs

A principal pair consists of a holomorphic principal $G$-bundle together with a holomorphic section of an associated Kaehler fibration. Such objects support natural gauge theoretic equations coming from a moment map condition, and also admit a notion of stability based on Geometric Invariant Theory. The Hitchin--Kobayashi correspondence for principal pairs identifies stability as the condition for the existence of solutions to the equations. In this paper we generalize these features in a way which allows the full gauge group of the principal bundle to be replaced by certain proper subgroups. Such a generalization is needed in order to use principal pairs as a general framework for describing augmented holomorphic bundles. We illustrate our results with applications to well known examples.

math.DG↗
arXiv2002

Surface group representations, Higgs bundles, and holomorphic triples

Using the $L^2$ norm of the Higgs field as a Morse function, we study the moduli spaces of $U(p,q)$-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on $(p,q)$. A key step is the identification of the function's local minima as moduli spaces of holomorphic triples. We prove that these moduli spaces of triples are irreducible and non-empty. Because of the relation between flat bundles and fundamental group representations, we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple $PU(p,q)$-representations. The topological invariants of the flat bundles bundle are used to label components. These invariants are bounded by a Milnor-Wood type inequality. For each allowed value of the invariants satisfying a certain coprimality condition, we prove that the corresponding component is non-empty and connected. If the coprimality condition does not hold, our results apply to the irreducible representations.

math.AG↗
arXiv2002

Coherent systems and Brill-Noether theory

Let $C$ be a curve of genus $g\geq 2$. A coherent system on $C$ consists of a pair $(E,V)$ where $E$ is an algebraic vector bundle of rank $n$ and degree $d$ and $V$ is a subspace of dimension $k$ of sections of $E$. The stability of the coherent systems depend on a parameter $τ$. We study the variation of the moduli space of coherent systems when we move the parameter. As an application, we analyse the cases $k=1,2,3$ and $n=2$ explicitly. For small values of $τ$, the moduli space of coherent systems is related to the Brill-Noether loci, the subspaces of the moduli space of stable bundles consisting of those bundles with a prescribed number of sections. The study of coherent systems is applied to find the dimension, irreducibility, and in some cases, the Picard group, of the Brill-Noether loci with $k\leq 3$.

math.AG↗
arXiv2001

Representations of the fundamental group of a surface in PU(p,q) and holomorphic triples

We count the connected components in the moduli space of PU(p,q)-representations of the fundamental group for a closed oriented surface. The components are labelled by pairs of integers which arise as topological invariants of the flat bundles associated to the representations. Our results show that for each allowed value of these invariants, which are bounded by a Milnor-Wood type inequality, there is a unique non-empty connected component. Interpreting the moduli space of representations as a moduli space of Higgs bundles, we take a Morse theoretic approach using a certain smooth proper function on the Higgs moduli space. A key step is the identification of the function's local minima as moduli spaces of holomorphic triples. We prove that these moduli spaces of triples are non-empty and irreducible.

math.AG↗