Constructible sheaves on toric varieties

TL;DR

This paper explicitly computes the category of constructible sheaves on toric varieties, extending known results over complex numbers to positive characteristic fields and providing new insights into étale sheaf theory and exodromy.

Contribution

It offers the first explicit computation of an étale exodromy theorem for stratifications over fields of positive characteristic with complex stratification structures.

Findings

Explicit description of constructible sheaves on toric varieties over complex numbers.

Extension of computations to split toric varieties over arbitrary fields.

First explicit étale exodromy theorem in positive characteristic for complex stratifications.

Abstract

This paper gives an explicit computation of the category of constructible sheaves on a toric variety (with respect to the stratification by torus orbits). Over the complex numbers, this simplifies a description due to Braden and Lunts. The same computation is carried out for split toric varieties over an arbitrary field, for constructible \'etale sheaves whose restriction to each stratum is locally constant and tamely ramified. This gives the first explicit computation of an \'etale exodromy theorem in positive characteristic for a stratification over a partially ordered set of height greater than 1.