Wild ramification, the nearby cycle complexes, and the characteristic cycles of $\ell$-adic sheaves

TL;DR

This paper establishes that the characteristic cycle of an étale sheaf near a point is determined solely by wild ramification at that point, refining previous results that required considering all boundary points.

Contribution

It proves a local, pointwise version of a known global result relating characteristic cycles to wild ramification, focusing on nearby cycle complexes.

Findings

Characteristic cycle determined by wild ramification at a point

Wild ramification of nearby cycle stalks is pointwise determined

Refines global ramification-characteristic cycle relationship

Abstract

We prove a purely local form of a result of Saito and Yatagawa. They proved that the characteristic cycle of a constructible \'etale sheaf is determined by wild ramification of the sheaf along the boundary of a compactification. But they had to consider ramification at all the points of the compactification. We give a pointwise result, that is, we prove that the characteristic cycle of a constructible \'etale sheaf around a point is determined by wild ramification at the point. The key ingredient is to prove that wild ramification of the stalk of the nearby cycle complex of a constructible \'etale sheaf at a point is determined by wild ramification at the point.