Trace and Kunneth formulas for singularity categories and applications

TL;DR

This paper develops trace and K"unneth formulas for singularity categories and inertia-invariant vanishing cycles, leading to a proof of a version of Bloch's Conductor Conjecture under specific monodromy conditions.

Contribution

It introduces an $ ext{l}$-adic trace formula for dg-categories and establishes K"unneth formulas for singularity categories, advancing the understanding of their structure and applications.

Findings

Proves an $ ext{l}$-adic trace formula for dg-categories.

Establishes K"unneth formulas for singularity categories.

Provides a proof of a version of Bloch's Conductor Conjecture with unipotent monodromy.

Abstract

We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal dg-category. Moreover, we prove K\"unneth formulas for dg-category of singularities, and for inertia-invariant vanishing cycles. As an application, we prove a version of Bloch's Conductor Conjecture (stated by Spencer Bloch in 1985), under the additional hypothesis that the monodromy action of the inertia group is unipotent.