Smooth varieties with torus actions

TL;DR

This paper characterizes smooth algebraic varieties with torus actions using combinatorial data, showing they are locally étale isomorphic to affine space with linear torus action, and provides a method to verify smoothness.

Contribution

It establishes a criterion for smoothness of torus-action varieties via combinatorial data and offers an effective way to check smoothness for Gm-threefolds.

Findings

Smooth varieties with torus actions are locally étale isomorphic to affine space with linear action.

The combinatorial data describing such varieties is locally isomorphic to that of affine space.

An effective method is provided to verify smoothness of Gm-threefolds using combinatorial data.

Abstract

In this paper we provide a characterization of smooth algebraic varieties endowed with a faithful algebraic torus action in terms of a combinatorial description given by Altmann and Hausen. Our main result is that such a variety X is smooth if and only if it is locally isomorphic in the \'etale topology to the affine space endowed with a linear torus action. Furthermore, this is the case if and only if the combinatorial data describing X is locally isomorphic in the \'etale topology to the combinatorial data describing affine space endowed with a linear torus action. Finally, we provide an effective method to check the smoothness of a Gm-threefold in terms of the combinatorial data.