On moduli spaces of Hitchin pairs

TL;DR

This paper investigates the geometric structure of moduli spaces of Hitchin pairs on Riemann surfaces, establishing conditions for irreducibility and demonstrating that these moduli spaces uniquely identify the underlying surface.

Contribution

It proves irreducibility of certain Hitchin moduli spaces and shows they uniquely determine the Riemann surface, linking geometric properties to surface classification.

Findings

Proved irreducibility of moduli spaces under specific conditions

Demonstrated the moduli space uniquely determines the Riemann surface

Established numerical conditions for the main results

Abstract

Let $X$ be a compact Riemann surface $X$ of genus at--least two. Fix a holomorphic line bundle $L$ over $X$. Let $\mathcal M$ be the moduli space of Hitchin pairs $(E ,\phi\in H^0(End(E)\otimes L))$ over $X$ of rank $r$ and fixed determinant of degree $d$. We prove that, for some numerical conditions, $\mathcal M$ is irreducible, and that the isomorphism class of the variety $\mathcal M$ uniquely determines the isomorphism class of the Riemann surface $X$.