Torus actions on affine varieties over characteristic zero fields

TL;DR

This paper extends the Altmann-Hausen presentation of affine varieties with torus actions from algebraically closed fields to arbitrary characteristic zero fields, incorporating Galois descent and torsor considerations.

Contribution

It generalizes the Altmann-Hausen framework to non-closed fields and introduces criteria for trivial torsors in two-dimensional torus actions.

Findings

Extended Altmann-Hausen presentation over arbitrary fields

Developed Galois descent techniques for affine varieties

Provided methods to identify trivial torsors in specific cases

Abstract

Using Galois descent tools, we extend the Altmann-Hausen presentation of normal affine algebraic varieties endowed with an effective torus action over an algebraically closed field of characteristic zero to the case where the ground field is an arbitrary field of characteristic zero. In this context, the acting torus may have non-trivial torsors and we need additional data to encode such varieties. Finally, we focus on affine varieties endowed with a two-dimensional torus action and we provide a method for determining when a torsor is trivial, in which case the Altmann-Hausen presentation simplifies.