Wild Ramification and the Cotangent Bundle

TL;DR

This paper introduces a new way to define the characteristic cycle of sheaves in positive characteristic, linking ramification, cotangent bundles, and intersection theory, with applications to Euler characteristics.

Contribution

It defines the characteristic cycle for sheaves with wild ramification, establishing compatibility with pull-back and local acyclicity, and relates it to characteristic cohomology classes.

Findings

Defined characteristic cycle in positive characteristic

Proved compatibility with pull-back and local acyclicity

Connected intersection with the 0-section to Euler-Poincaré characteristic

Abstract

We define the characteristic cycle of a locally constant \'etale sheaf on a smooth variety in positive characteristic ramified along boundary as a cycle in the cotangent bundle of the variety, at least on a neighborhood of the generic point of the divisor on the boundary. The crucial ingredient in the definition is an additive structure on the boundary induced by the groupoid structure of multiple self products. We prove a compatibility with pull-back and local acyclicity in non-characteristic situations. We also give a relation with the characteristic cohomology class under a certain condition and a concrete example where the intersection with the 0-section computes the Euler-Poincar\'e characteristic.