Divisorial fans and algebraic torus actions over arbitrary fields

TL;DR

The paper develops a comprehensive framework to descend normal $T$-varieties over arbitrary fields by encoding them as divisorial fans on a normal semiprojective base, equipped with finite Galois (semilinear) actions. It introduces a localized category of pp-divisors to establish an equivalence with the category of affine $T$-varieties, enabling Galois descent and a unified treatment across all complexities. The main theorem generalizes previous results by Huruguen and AHS08, showing that normal $T$-varieties with a torus action over $k$ arise from divisorial fans with a compatible Galois action, provided certain quasi-projectivity conditions hold on Galois orbits. The complexity-one case yields a more tractable descent criterion and yields concrete applications to explicit varieties such as Hirzebruch surfaces and projective spaces, illustrating how $k$-forms can be toric or non-toric depending on the Galois data. Overall, the work provides a powerful combinatorial toolkit for torus actions over arbitrary fields with broad implications for the study of $T$-varieties and their forms.

Abstract

We provide a algebro-geometric combinatorial description of geometrically integral geometrically normal varieties endowed with an effective action of an algebraic torus over arbitrary fields. This description is achieved in terms of divisorial fans endowed with a Galois semilinear action. This work concludes the description of normal $T$-varieties over fields.

Theorems & Definitions (144)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3