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Equivariant Hodge modules and rational singularities

Donu Arapura, Scott Hiatt

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.

Equivariant Hodge modules and rational singularities

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 is an equivariant Hodge module with rational singularities, then the induced module on the GIT quotient also has rational singularities.
Paper Structure (3 sections, 23 theorems, 52 equations)

This paper contains 3 sections, 23 theorems, 52 equations.

Table of Contents

  1. Rational Singularities for Pure Hodge Modules
  2. Invariant theory
  3. Equivariant Hodge modules

Key Result

Proposition 1

Suppose that $G$ is a reductive group that acts effectively on $X$ such that there exists a stable point. Let $Y = X//G$. Then there is an equivalence of categories between $HM_{Y}(Y,w)$ and $HM_{X,G}(X, w+d)$ .

Theorems & Definitions (42)

  • Proposition 1
  • Theorem 2
  • Corollary 3
  • Definition 1.2
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Corollary 1.5
  • proof
  • ...and 32 more