Wild ramification and the cotangent bundle

Abstract

We define the characteristic cycle of a locally constant étale sheaf on a smooth variety in positive characteristic ramified along the 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 the commutative group 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 cohomological characteristic class under a certain condition and a concrete example where the intersection with the 0 0 -section computes the Euler-Poincaré characteristic.