Rational singularities, quiver moment maps, and representations of surface groups

TL;DR

This paper proves that certain algebraic varieties related to quiver moment maps and surface group representations have rational singularities, impacting the understanding of their geometric and arithmetic properties.

Contribution

It establishes the rational singularities of zero loci of quiver moment maps and representation spaces of surface groups using jet schemes, a novel approach in this context.

Findings

Zero loci of quiver moment maps have rational singularities.

Representation spaces of surface groups have rational singularities.

Implications for the representation zeta function of SL over integers and p-adic integers.

Abstract

We prove using jet schemes that the zero loci of the moment maps for the quivers with one vertex and at least two loops have rational singularities. This implies that the spaces of representations of the fundamental group of a compact Riemann surface of genus at least two have rational singularities. This has consequences for the representation zeta function of the special linear group over the integers and over the p-adic integers.