\'Etale Splittings of Certain Azumaya Algebras on Toric and Hypertoric Varieties in Positive Characteristic

TL;DR

This paper constructs an étale cover for toric varieties in positive characteristic to trivialize sheaves of differential operators, extends this to hypertoric varieties, and discusses implications for localization theorems.

Contribution

It introduces a method to trivialize Azumaya algebras on toric and hypertoric varieties in positive characteristic, advancing understanding of differential operators in this setting.

Findings

Constructed étale cover trivializing sheaves of differential operators.

Extended trivialization to Azumaya algebras on hypertoric varieties.

Formulated criteria for derived Beilinson-Bernstein localization.

Abstract

For a smooth toric variety X over a field of positive characteristic, a T-equivariant \'{e}tale cover Y \rightarrow T^*X^{(1)} trivializing the sheaf of crystalline differential operators on X is constructed. This trivialization is used to show that the sheaf of differential operators is a trivial Azumaya algebra along the fibers of the moment map. This result is then extended to certain Azumaya algebras on hypertoric varieties, whose global sections are central reductions of the hypertoric enveloping algebra in positive characteristic. A criteria for a derived Beilinson-Bernstein localization theorem is then formulated.