Symplectic resolutions for Higgs moduli spaces

TL;DR

This paper investigates the algebraic symplectic geometry of Higgs moduli spaces, establishing conditions under which these singular spaces admit symplectic resolutions, with implications for their geometric structure.

Contribution

It proves that Higgs moduli spaces are symplectic singularities and characterizes when they admit projective symplectic resolutions, extending understanding of their geometric properties.

Findings

Higgs moduli spaces are symplectic singularities.

They admit projective symplectic resolutions only when g=1 or (g,n)=(2,2).

Results rely on the Isosingularity Theorem and recent advances in the field.

Abstract

In this paper, we study the algebraic symplectic geometry of the singular moduli spaces of Higgs bundles of degree $0$ and rank $n$ on a compact Riemann surface $X$ of genus $g$. In particular, we prove that such moduli spaces are symplectic singularities, in the sense of Beauville [Bea00], and admit a projective symplectic resolution if and only if $g=1$ or $(g, n)=(2,2)$. These results are an application of a recent paper by Bellamy and Schedler [BS16] via the so-called Isosingularity Theorem.