Cohomological Milnor formula and Saito's conjecture on characteristic classes

TL;DR

This paper proves Saito's conjecture relating cohomological characteristic classes to characteristic cycles, and develops cohomological Milnor and conductor formulas for constructible sheaves on possibly singular varieties.

Contribution

It confirms Saito's conjecture in the quasi-projective case and introduces a new cohomological characteristic class with applications to Milnor and conductor formulas.

Findings

Confirmed Saito's conjecture for quasi-projective varieties.

Constructed a cohomological characteristic class supported on non-acyclicity locus.

Proved cohomological Milnor and conductor formulas for constructible sheaves.

Abstract

We confirm the quasi-projective case of Saito's conjecture, namely that the cohomological characteristic classes defined by Abbes and Saito can be computed in terms of the characteristic cycles. We construct a cohomological characteristic class supported on the non-acyclicity locus of a separated morphism relatively to a constructible sheaf. As applications of the functorial properties of this class, we prove cohomological analogs of the Milnor formula and the conductor formula for constructible sheaves on (not necessarily smooth) varieties.