Proper polyhedral divisors and torus actions over arbitrary fields

TL;DR

This work extends the Altmann–Hausen framework for describing affine normal varieties with torus actions to arbitrary fields by incorporating Galois-decent data into proper polyhedral divisors. The core idea is to use pp-divisors equipped with a Galois semilinear action to capture the torus action over a base field and to recover the ground-field structure via descent, including a precise correspondence between geometrically integral affine k-varieties with split-torus actions and pairs $(\mathfrak{D},g)$ in $\mathfrak{PPDiv}(\Gamma)$. The paper develops a robust functorial picture: pp-divisors form a category; the Altmann–Hausen construction gives a covariant functor to $T$-varieties; semilinear morphisms/equivariant morphisms model Galois descent, and base-change results ensure stability under extension. A main outcome is that nonsplit affine normal $T$-varieties over $k$ arise from minimal pp-divisors on geometrically integral semiprojective bases, with descent data encoding all $k$-forms; the framework is then illustrated through applications to minimal pp-divisors and the relationship to the “other” associated $T$-variety. Overall, the paper provides a comprehensive algebraic-geometric toolkit for torus actions over arbitrary fields, unifying descent theory, convex-geometric data, and polyhedral divisors in a categorical setting.

Abstract

We provide a algebro-geometric combinatorial description of geometrically integral geometrically normal affine varieties endowed with an effective action of an algebraic torus over arbitrary fields. This description is achieved in terms of proper polyhedral divisors endowed with a Galois semilinear action.

Theorems & Definitions (147)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Proposition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4