An Analytic Approach to the Quasi-projectivity of the Moduli Space of Higgs Bundles

TL;DR

This paper employs analytic techniques, including symplectic cuts, to demonstrate that the moduli space of Higgs bundles is quasi-projective without assuming smoothness, extending previous results by Hausel.

Contribution

It introduces an analytic approach to prove the quasi-projectivity of Higgs bundle moduli spaces without requiring smoothness, using symplectic cuts for compactification.

Findings

Established the quasi-projectivity of the moduli space

Constructed a normal, projective compactification

Extended Hausel's method without smoothness assumption

Abstract

The moduli space of Higgs bundles can be defined as a quotient of an infinite-dimensional space. Moreover, by the Kuranishi slice method, it is equipped with the structure of a normal complex space. In this paper, we will use analytic methods to show that the moduli space is quasi-projective. In fact, following Hausel's method, we will use the symplectic cut to construct a normal and projective compactification of the moduli space, and hence prove the quasi-projectivity. The main difference between this paper and Hausel's is that the smoothness of the moduli space is not assumed.