Equivariant Hodge modules and rational singularities

TL;DR

This paper introduces a new concept of Hodge modules with rational singularities and extends Boutot's theorem to show that certain GIT quotients inherit rational singularities under group actions.

Contribution

It defines equivariant Hodge modules with rational singularities and generalizes Boutot's theorem to these modules in the context of GIT quotients.

Findings

Hodge modules with rational singularities are well-defined.

GIT quotients inherit rational singularities under certain conditions.

Extension of Boutot's theorem to equivariant Hodge modules.

Abstract

We define a notion of Hodge modules with rational singularities. A variety has rational singularities in the usual sense, if it is normal and the Hodge module related to intersection cohomology has rational singularities in the present sense. Our main result is a generalization of Boutot's theorem that if a reductive group acts on an affine variety with a stable point, and $H$ is an equivariant Hodge module with rational singularities, then the induced module on the GIT quotient also has rational singularities.