Picard group and fundamental group of the moduli of Higgs bundles on curves

TL;DR

This paper computes the fundamental and Picard groups of the moduli space of semistable principal G-Higgs bundles on a smooth projective curve of genus at least 2, advancing understanding of their geometric and topological properties.

Contribution

It provides the first explicit calculations of the fundamental and Picard groups for these moduli spaces, revealing their structure and topological invariants.

Findings

Fundamental group of the moduli space is computed.

Picard group of the moduli space is determined.

Results apply to a broad class of reductive groups.

Abstract

Let $X$ be an irreducible smooth projective curve of genus $g \geq 2$ over $\mathbb{C}$. Let $G$ be a connected reductive affine algebraic group over $\mathbb{C}$. Let $\mathrm{M}_{G, {\rm Higgs}}^{\delta}$ be the moduli space of semistable principal $G$--Higgs bundles on $X$ of topological type $\delta \in \pi_1(G)$. In this article, we compute the fundamental group and Picard group of $\mathrm{M}_{G, {\rm Higgs}}^{\delta}$.