Cohomological Milnor formula and Saito's conjecture on characteristic classes

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.

arXiv commentFinal version. Note that the numbering of sections, propositions, and formulas in the arXiv version is different from that in the journal version

References in corpus (6)

• The characteristic cycle and the singular support of a constructible sheaf
• The characteristic class and ramification of an l-adic etale sheaf
• Categorical traces and a relative Lefschetz-Verdier formula
• Characteristic cycle of the exterior product of constructible sheaves
• Characteristic cycles and the conductor of direct image
• Cotangent bundle and micro-supports in mixed characteristic case