On the stability of vanishing cycles of \'etale sheaves in positive characteristic

TL;DR

This paper investigates the stability of vanishing cycles of étale sheaves in positive characteristic, revealing that on smooth surfaces, their dependence on test functions is limited to finite jets, with conjectures extending this to higher dimensions.

Contribution

It demonstrates finite jet dependence of vanishing cycles on smooth surfaces and identifies a class of stable sheaves, including tame simple normal crossing sheaves, stable under Radon transform.

Findings

Vanishing cycles depend only on finite jets of test functions on smooth surfaces.

Tame simple normal crossing sheaves exhibit strong stability of vanishing cycles.

Stability of this class persists under Radon transform.

Abstract

In positive characteristic, in contrast to the complex analytic case, vanishing cycles are highly sensitive to test functions (the maps to the henselian traits). We study this dependence and show that on a smooth surface, this dependence is generically only up to a finite jet of the test functions. We conjecture that this continues to hold in higher dimensions. We also study the class of sheaves whose vanishing cycles have the strongest stability. Among other things, we show that tame simple normal crossing sheaves belong to this class, and this class is stable under the Radon transform.