arXiv subjects

Oscar Garcia-Prada

Publications and source records attributed to Oscar Garcia-Prada.

At least 19 recordsLinked to original sources

arXiv2024

Multiplicative Higgs bundles and involutions

In this paper we generalize the theory of multiplicative $G$-Higgs bundles over a curve to pairs $(G,θ)$, where $G$ is a reductive algebraic group and $θ$ is an involution of $G$. This generalization involves the notion of a multiplicative Higgs bundle taking values in a symmetric variety associated to $θ$, or in an equivariant embedding of it. We also study how these objects appear as fixed points of involutions of the moduli space of multiplicative $G$-Higgs bundles, induced by the involution $θ$.

math.AG↗

arXiv2024

A general Cayley correspondence and higher Teichmüller spaces

We introduce a new class of $\mathfrak{sl}_2$-triples in a complex simple Lie algebra $\mathfrak{g}$, which we call magical. Such an $\mathfrak{sl}_2$-triple canonically defines a real form and various decompositions of $\mathfrak{g}$. Using this decomposition data, we explicitly parameterize special connected components of the moduli space of Higgs bundles on a compact Riemann surface $X$ for an associated real Lie group, hence also of the corresponding character variety of representations of $π_1X$ in the associated real Lie group. This recovers known components when the real group is split, Hermitian of tube type, or $\mathrm{SO}_{p,q}$ with $1<p\leq q$, and also constructs previously unknown components for the quaternionic real forms of $\mathrm{E}_6$, $\mathrm{E}_7$, $\mathrm{E}_8$ and $\mathrm{F}_4$. The classification of magical $\mathfrak{sl}_2$-triples is shown to be in bijection with the set of $Θ$-positive structures in the sense of Guichard--Wienhard, thus the mentioned parameterization conjecturally detects all examples of higher Teichmüller spaces. Indeed, we discuss properties of the surface group representations obtained from these Higgs bundle components and their relation to $Θ$-positive Anosov representations, which indicate that this conjecture holds.

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arXiv2023

Higgs bundles twisted by a vector bundle

In this paper, we consider a generalization of the theory of Higgs bundles over a smooth complex projective curve in which the twisting of the Higgs field by the canonical bundle of the curve is replaced by a rank 2 vector bundle. We define a Hitchin map and give a spectral correspondence. We also state a Hitchin-Kobayashi correspondence for a generalization of the Hitchin equations to this situation. In a certain sense, this theory lies halfway between the theories of Higgs bundles on a curve and on a higher dimensional variety.

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arXiv2021

Complex structures, moment maps, and the Ricci form (Extended Version)

The Ricci form is a moment map for the action of the group of exact volume preserving diffeomorphisms on the space of almost complex structures. This observation yields a new approach to the Weil-Petersson symplectic form on the Teichmuller space of isotopy classes of complex structures with real first Chern class zero and nonempty Kahler cone.

math.SG↗

arXiv2021

Arakelov-Milnor inequalities and maximal variations of Hodge structure

In this paper we study the $\mathbb{C}^*$-fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to complex variations of Hodge structure. We introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups. A main result of this paper is a bound on this invariant which generalizes both the Milnor-Wood inequality of the Hermitian case and the Arakelov inequalities of classical variations of Hodge structure. When the generalized Toledo invariant is maximal, we establish rigidity results for the associated variations of Hodge structure which generalize known rigidity results for maximal Higgs bundles and their associated maximal representations in the Hermitian case.

math.AG↗

arXiv2020

Real Higgs pairs and Non-abelian Hodge correspondence on a Klein surface

We introduce real structures on $L$-twisted Higgs pairs over a compact Riemann surface equipped with an anti-holomorphic involution, and prove a Hitchin--Kobayashi correspondence for them. Real $G$-Higgs bundles, where $G$ is a real form of a connected semisimple complex affine algebraic group $G^{\mathbb{C}}$, constitute a particular class of examples of these pairs. The real structure in this case involves a conjugation of $G^{\mathbb{C}}$ commuting with the one defining the real form $G$. We establish a homeomorphism between the moduli space of real $G$-Higgs bundles and the moduli space of compatible representations of the orbifold fundamental group of $X$. Finally, we show how real $G$-Higgs bundles appear naturally as fixed points of certain anti-holomorphic involutions of the moduli space of $G$-Higgs bundles, that are constructed using the real structures on $G^{\mathbb{C}}$ and $X$.

math.DG↗

arXiv2020

Action of the mapping class group on character varieties and Higgs bundles

We consider the action of a finite subgroup of the mapping class group $Mod(S)$ of an oriented compact surface $S$ of genus $g \geq 2$ on the moduli space $\mathcal{R}(S,G)$ of representations of $π_1(S)$ in a connected semisimple real Lie group $G$. Kerckhoff's solution of the Nielsen realization problem ensures the existence of an element $J$ in the Teichmüller space of $S$ for which $Γ$ can be realised as a subgroup of the group of automorphisms of $X=(S,J)$ which are holomorphic or antiholomorphic. We identify the fixed points of the action of $Γ$ on $\mathcal{R}(S,G)$ in terms of $G$-Higgs bundles on $X$ equipped with a certain twisted $Γ$-equivariant structure, where the twisting involves abelian and non-abelian group cohomology simultaneously. When the kernel of the isotropy representation of the maximal compact subgroup of $G$ is trivial, the fixed points can be described in terms of familiar objects on $Y=X/Γ^+$, where $Γ^+ \subset Γ$ is the maximal subgroup of $Γ$ consisting of holomorphic automorphisms of $X$. If $Γ=Γ^+$ one obtains actual $Γ$-equivariant $G$-Higgs bundles on $X$, which in turn correspond with parabolic Higgs bundles on $Y=X/Γ$ (this generalizes work of Nasatyr \& Steer for $G=SL(2,\mathbb{R})$ and Boden, Andersen & Grove and Furuta & Steer for $G=SU(n)$). If on the other hand $Γ$ has antiholomorphic automorphisms, the objects on $Y=X/Γ^+$ correspond with pseudoreal parabolic Higgs bundles. This is a generalization in the parabolic setup of the pseudoreal Higgs bundles studied by the first author in collaboration with Biswas & Hurtubise.

math.AG↗

arXiv2020

A moment map interpretation of the Ricci form, Kähler--Einstein structures, and Teichmüller spaces

This paper surveys the role of moment maps in Kähler geometry. The first section discusses the Ricci form as a moment map and then moves on to moment map interpretations of the Kähler--Einstein condition and the scalar curvature (Quillen--Fujiki--Donaldson). The second section examines the ramifications of these results for various Teichmüller spaces and their Weil--Petersson symplectic forms and explains how these arise naturally from the construction of symplectic quotients. The third section discusses a symplectic form introduced by Donaldson on the space of Fano complex structures.

math.SG↗

arXiv2019

Parabolic Higgs bundles and representations of the fundamental group of a punctured surface into a real group

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. Three interesting features are the relation between the parabolic degree and the geometry of the Tits boundary, the treatment of the case when the logarithm of the monodromy is on the boundary of a Weyl alcove, and the correspondence of the orbits encoding the singularity via the Kostant-Sekiguchi correspondence. We also describe some special features of the moduli spaces when G is a split real form or a group of Hermitian type.

math.DG↗

arXiv2019

Involutions and higher order automorphisms of Higgs bundle moduli spaces

We consider the moduli space $\mathcal{M}(G)$ of $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a complex semisimple Lie group. This is a hyperkähler manifold homeomorphic to the moduli space $\mathcal{R}(G)$ of representations of the fundamental group of $X$ in $G$. In this paper we study finite order automorphisms of $\mathcal{M}(G)$ obtained by combining the action of an element of order $n$ in $H^1(X,Z)\rtimes \mbox{Out}(G)$, where $Z$ is the centre of $G$ and $\mbox{Out}(G)$ is the group of outer automorphisms of $G$, with the multiplication of the Higgs field by an $n$th-root of unity, and describe the subvarieties of fixed points. We give special attention to the case of involutions, defined by the action of an element of order $2$ in $H^1(X,Z)\rtimes\mbox{Out}(G)$ combined with the multiplication of the Higgs field by $\pm 1$. In this situation, the subvarieties of fixed points are hyperkähler submanifolds of $\mathcal{M}(G)$ in the (+1)-case, corresponding to the moduli space of representations of the fundamental group in certain reductive complex subgroups of $G$ defined by holomorphic involutions of $G$; while in the (-1)-case they are Lagrangian subvarieties corresponding to the moduli space of representations of the fundamental group of $X$ in real forms of $G$ and certain extensions of these. We illustrate the general theory with the description of involutions for $G=\mbox{SL}(n,\mathbb{C})$ and involutions and order three automorphism defined by triality for $G=\mbox{Spin}(8,\mathbb{C})$.

math.AG↗

arXiv2018

Irreducibility of moduli of semistable Chains and applications to U(p,q)-Higgs bundles

We give necessary and sufficient conditions for moduli spaces of semistable chains on a curve to be irreducible and non-empty. This gives information on the irreducible components of the nilpotent cone of GL_n-Higgs bundles and the irreducible components of moduli of systems of Hodge bundles on curves. As we do not impose coprimality restrictions, we can apply this to prove connectedness for moduli spaces of U(p,q)-Higgs bundles.

math.AG↗

arXiv2018

Exotic components of $\mathrm{SO}(p,q)$ surface group representations, and their Higgs bundle avatars

For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups $\mathrm{SO}(p,q)$ with $2<p<q$. These groups lie outside formerly know classes of groups associated with exotic components.

math.AG↗

arXiv2018

Higgs bundles over elliptic curves for complex reductive Lie groups

We study Higgs bundles over an elliptic curve with complex reductive structure group, describing the (normalization of) its moduli spaces and the associated Hitchin fibration. The case of trivial degree is covered by the work of Thaddeus in 2001. Our arguments are different from those of Thaddeus and cover arbitrary degree.

math.AG↗

arXiv2018

Involutions of rank 2 Higgs bundle moduli spaces

We consider the moduli space of rank 2 Higgs bundles with fixed determinant over a smooth projective curve X of genus 2 over the complex numbers, and study involutions defined by tensoring the vector bundle with an element $α$ of order 2 in the Jacobian of the curve, combined with multiplication of the Higgs field by $\pm 1$. We describe the fixed points of these involutions in terms of the Prym variety of the covering of $X$ defined by $α$, and give an interpretation in terms of the moduli space of representations of the fundamental group.

math.AG↗

arXiv2017

Higgs bundles, the Toledo invariant and the Cayley correspondence

We define the Toledo invariant of a G-Higgs bundle on a Riemann surface, where G is a real semisimple group of Hermitian type, and we prove a Milnor-Wood type bound for this invariant when the bundle is semistable. We prove rigidity results when the Toledo invariant is maximal, establishing in particular a Cayley correspondence when the symmetric space defined by G is of tube type. This gives a new proof of the Milnor-Wood inequality of Burger-Iozzi-Wienhard for representations of the fundamental group of a Riemann surface into G. Compared to previous results using Higgs bundles, it uses general theory and avoids any case by case study.

math.DG↗

arXiv2015

Pseudo-real principal $G$-bundles over a real curve

We consider stable and semistable principal bundles over a smooth projective real algebraic curve, equipped with a real or pseudo-real structure in the sense of Atiyah. After fixing suitable topological invariants, one can build a suitable gauge theory, and show that the resulting moduli spaces of pseudo-real bundles are connected. This in turn allows one to describe the various fixed point varieties on the complex moduli spaces under the action of the real involutions on the curve and the structure group.

math.AG↗

arXiv2014

Higgs bundles for the non-compact dual of the special orthogonal group

Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group G. In this paper we examine the case G=SO*(2n). We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli space is connected in this maximal Toledo case. The Morse theory also allows us to show connectedness when the Toledo invariant is zero. The correspondence between Higgs bundles and surface group representations thus allows us to count the connected components with zero and maximal Toledo invariant in the moduli space of representations of the fundamental group of the surface in SO*(2n).

math.AG↗

arXiv2013

Coupled equations for Kähler metrics and Yang-Mills connections

We study equations on a principal bundle over a compact complex manifold coupling a connection on the bundle with a Kahler structure on the base. These equations generalize the conditions of constant scalar curvature for a Kahler metric and Hermite-Yang-Mills for a connection. We provide a moment map interpretation of the equations and study obstructions for the existence of solutions, generalizing the Futaki invariant, the Mabuchi K-energy and geodesic stability. We finish by giving some examples of solutions.

math.DG↗