Birationality of moduli spaces of twisted $\mathrm{U}(p,q)$-Higgs bundles

TL;DR

This paper investigates the birational relationships of moduli spaces of twisted U(p,q)-Higgs bundles on Riemann surfaces, revealing their structure changes with a parameter and establishing irreducibility through wall crossing analysis.

Contribution

It demonstrates that moduli spaces of twisted U(p,q)-Higgs bundles are birational within certain parameter ranges and applies quiver bundle techniques and Hitchin-Kobayashi correspondence to analyze their properties.

Findings

Moduli spaces are birational for specific parameter ranges.

Wall crossing affects the structure of moduli spaces.

Irreducibility results are obtained using known Higgs bundle theories.

Abstract

A $\mathrm{U}(p,q)$-Higgs bundle on a Riemann surface (twisted by a line bundle) consists of a pair of holomorphic vector bundles, together with a pair of (twisted) maps between them. Their moduli spaces depend on a real parameter $\alpha$. In this paper we study wall crossing for the moduli spaces of $\alpha$-polystable twisted $\mathrm{U}(p,q)$-Higgs bundles. Our main result is that the moduli spaces are birational for a certain range of the parameter and we deduce irreducibility results using known results on Higgs bundles. Quiver bundles and the Hitchin-Kobayashi correspondence play an essential role.