Applications of perverse sheaves in commutative algebra

TL;DR

This paper explores how perverse sheaves relate to singularity properties and local cohomology in algebraic geometry, developing a theory of perverse $ extbf{F}_p$-sheaves and extending vanishing theorems.

Contribution

It introduces a theory of perverse $ extbf{F}_p$-sheaves in characteristic $p$ and connects their properties to singularity invariants via Riemann-Hilbert correspondences.

Findings

Development of a theory of perverse $ extbf{F}_p$-sheaves

Extension of the Artin vanishing theorem in characteristic $p$

Connections between perverse sheaves and singularity properties

Abstract

The goal of this paper is to explain how basic properties of perverse sheaves sometimes translate via Riemann-Hilbert correspondences (in both characteristic $0$ and characteristic $p$) to highly non-trivial properties of singularities, especially their local cohomology. Along the way, we develop a theory of perverse $\mathbf{F}_p$-sheaves on varieties in characteristic $p$, expanding on previous work by various authors, and including a strong version of the Artin vanishing theorem.