Cohen-Macaulay Du Bois singularities with a torus action of complexity one

TL;DR

This paper characterizes Cohen-Macaulay Du Bois singularities with a torus action of complexity one using divisorial fans and cohomological criteria, providing new examples that lack rational singularities.

Contribution

It introduces cohomological criteria for Cohen-Macaulay Du Bois $ ext{T}$-varieties of complexity one using polyhedral divisors and provides novel examples without rational singularities.

Findings

Cohomological criteria for Cohen-Macaulay Du Bois $ ext{T}$-varieties

Example of a Cohen-Macaulay Du Bois singularity without rational singularities

Application of divisorial fan description to singularity analysis

Abstract

Using Altmann-Hausen-Suss description of $\mathbb{T}$-varieties via divisorial fans and K\'ovacs-Schwede-Smith characterization of Du Bois singularities, we study Cohen-Macaulay Du Bois $\mathbb{T}$-singularities of complexity one. We exhibit cohomological criteria for a $\mathbb{T}$-variety to be Cohen-Macaulay and Du Bois in terms of polyhedral divisors. We give an example of a Cohen-Macaulay Du Bois singularity of complexity one which does not have rational singularities.