Rational singularities of normal T-varieties

TL;DR

This paper characterizes when normal T-varieties have rational singularities by computing higher direct images of desingularizations using combinatorial data, and explores related Cohen-Macaulay properties and specific surface singularities.

Contribution

It provides a combinatorial criterion for rational singularities of T-varieties and extends understanding of their Cohen-Macaulay conditions using polyhedral divisors.

Findings

A criterion for rational singularities of T-varieties.

Partial criterion for Cohen-Macaulay T-varieties.

Characterization of quasihomogeneous elliptic surface singularities.

Abstract

A T-variety is an algebraic variety X with an effective regular action of an algebraic torus T. Altmann and Hausen gave a combinatorial description of an affine T-variety X by means of polyhedral divisors. In this paper we compute the higher direct images of the structure sheaf of a desingularization of X in terms of this combinatorial data. As a consequence, we give a criterion as to when a T-variety has rational singularities. We also provide a partial criterion for a T-variety to be Cohen-Macaulay. As an application we characterize in this terms quasihomogeneous elliptic singularities of surfaces.