On the relative twist formula of $\ell$-adic sheaves

TL;DR

This paper proposes a conjecture on the relative twist formula for $$-adic sheaves, verifies it under certain conditions, and explores its relation to a new cohomological characteristic class.

Contribution

It introduces a conjecture generalizing Kato-Saito's work, defines a relative cohomological characteristic class, and establishes its compatibility with proper push-forward.

Findings

Verification of the conjecture under transversal assumptions

Definition of a relative cohomological characteristic class

Compatibility of the class formation with proper push-forward

Abstract

We propose a conjecture on the relative twist formula of $\ell$-adic sheaves, which can be viewed as a generalization of Kato-Saito's conjecture. We verify this conjecture under some transversal assumptions. We also define a relative cohomological characteristic class and prove that its formation is compatible with proper push-forward. A conjectural relation is also given between the relative twist formula and the relative cohomological characteristic class.