Moduli of Parabolic Higgs Bundles and Atiyah Algebroids

TL;DR

This paper explores the geometric structure of moduli spaces of parabolic Higgs bundles, revealing their Poisson geometry, integrability, and connections to Atiyah algebroids, with implications for Hitchin systems and algebraic geometry.

Contribution

It establishes a Poisson structure on moduli spaces of parabolic Higgs bundles and links them to Atiyah algebroids, extending known resolutions and integrability results.

Findings

Moduli spaces possess a natural Poisson structure.

Full flag cases yield a Grothendieck-Springer resolution.

All studied moduli spaces are integrable systems.

Abstract

In this paper we study the geometry of the moduli space of (non-strongly) parabolic Higgs bundles over a Riemann surface with marked points. We show that this space possesses a Poisson structure, extending the one on the dual of an Atiyah algebroid over the moduli space of parabolic vector bundles. By considering the case of full flags, we get a Grothendieck-Springer resolution for all other flag types, in particular for the moduli spaces of twisted Higgs bundles, as studied by Markman and Bottacin and used in the recent work of Laumon-Ng\^o. We discuss the Hitchin system, and demonstrate that all these moduli spaces are integrable systems in the Poisson sense.