The characteristic cycle and the singular support of a constructible sheaf

TL;DR

This paper introduces the characteristic cycle of an etale sheaf in positive characteristic, linking it to singular support, and generalizes classical formulas like Grothendieck-Ogg-Shafarevich to higher dimensions.

Contribution

It defines the characteristic cycle using singular support in positive characteristic and proves a Milnor-type formula and an index formula for higher-dimensional varieties.

Findings

Established a formula for the total dimension of vanishing cycles

Derived an index formula for Euler-Poincare characteristic

Generalized semi-continuity of the Swan conductor to higher dimensions

Abstract

We define the characteristic cycle of an etale sheaf as a cycle on the cotangent bundle of a smooth variety in positive characteristic using the singular support recently defined by Beilinson. We prove a formula a la Milnor for the total dimension of the space of vanishing cycles and an index formula computing the Euler-Poincare characteristic, generalizing the Grothendieck-Ogg-Shafarevich formula to higher dimension. An essential ingredient of the construction and the proof is a partial generalization to higher dimension of the semi-continuity of the Swan conductor due to Deligne-Laumon. We prove the index formula by establishing certain functorial properties of characteristic cycles.