Rational singularities and quotients by holomorphic group actions

TL;DR

This paper proves that rational singularities are preserved under quotients by holomorphic group actions, extending Boutot's algebraic results to the analytic setting and providing new vanishing theorems.

Contribution

It extends the stability of rational singularities under group quotients from algebraic to complex analytic spaces and introduces new cohomology vanishing theorems.

Findings

Rational singularities are stable under holomorphic quotients.

Extension of Boutot's result to the analytic category.

New vanishing theorems for cohomology groups.

Abstract

We prove that rational and 1-rational singularities of complex spaces are stable under taking quotients by holomorphic actions of reductive and compact Lie groups. This extends a result of Boutot to the analytic category and yields a refinement of his result in the algebraic category. As one of the main technical tools vanishing theorems for cohomology groups with support on fibres of resolutions are proven.