arXiv subjects

O. García-Prada

Publications and source records attributed to O. García-Prada.

4 recordsLinked to original sources

arXiv2025

A Prym-Narasimhan-Ramanan construction of principal bundle fixed points

Let $X$ be a compact Riemann surface and $G$ be a connected reductive complex Lie group with centre $Z$. Consider the moduli space $M(X,G)$ of polystable principal holomorphic $G$-bundles on $X$. There is an action of the group $H^1(X,Z)$ of isomorphism classes of $Z$-bundles over $X$ on $M(X,G)$ induced by the multiplication $Z\times G\to G.$ Let $Γ$ be a finite subgroup of $H^1(X,Z)$. Our goal is to find a Prym--Narasimhan--Ramanan-type construction to describe the fixed points of $M(X,G)$ under the action of $Γ$. A main ingredient in this construction is the theory of twisted equivariant bundles on an étale cover of $X$ developed in arXiv:2208.0902(2).

math.AG↗

arXiv2024

Gravitating vortices and Symplectic Reduction by Stages

We undertake a novel approach to the existence problem for gravitating vortices on a Riemann surface based on symplectic reduction by stages, which seems to be new in the PDE as well as the gauge theory literature. The main technical tool for our study is the reduced $α$-K-energy, for which we establish convexity properties by means of finite-energy pluripotential theory, as recently applied to the study of constant scalar curvature Kähler metrics. Using these methods, we prove that the existence of solutions to the gravitating vortex equations on the sphere implies the polystability of the effective divisor defined by the zeroes of the Higgs field. This approach also enables us to establish the uniqueness of gravitating vortices in any admissible Kähler class, in the absence of automorphisms. Lastly, we also prove the existence of solutions for the gravitating vortex equations for genus $g\geq 1$ for certain ranges of the coupling constant $α$ and the volume.

math.DG↗

arXiv2022

Non-connected Lie groups, twisted equivariant bundles and coverings

Let $Γ$ be a finite group acting on a Lie group $G$. We consider a class of group extensions $1 \to G \to \hat{G} \to Γ\to 1$ defined by this action and a $2$-cocycle of $Γ$ with values in the centre of $G$. We establish and study a correspondence between $\hat{G}$-bundles on a manifold and twisted $Γ$-equivariant bundles with structure group $G$ on a suitable Galois $Γ$-covering of the manifold. We also describe this correspondence in terms of non-abelian cohomology. Our results apply, in particular, to the case of a compact or reductive complex Lie group $\hat{G}$, since such a group is always isomorphic to an extension as above, where $G$ is the connected component of the identity and $Γ$ is the group of connected components of $\hat{G}$.

math.DG↗

arXiv2005

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

We compute the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, 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 study their variation with this parameter.

math.AG↗