Wild Ramification and Restrictions to Curves

TL;DR

This paper demonstrates that wild ramification properties of constructible sheaves on higher-dimensional varieties are fully determined by their restrictions to curves, impacting the understanding of Euler characteristics and Swan conductors.

Contribution

It establishes that wild ramification on surfaces and higher dimensions can be understood through restrictions to curves, providing new insights into ramification invariants.

Findings

Wild ramification on surfaces is determined by restrictions to curves.

Euler-Poincaré characteristic is determined by ramification on curves.

Sum of Swan conductors is also determined by restrictions to curves.

Abstract

We prove that wild ramification of a constructible sheaf on a surface is determined by that of the restrictions to all curves. We deduce from this result that the Euler-Poincar\'e characteristic of a constructible sheaf on a variety of arbitrary dimension over an algebraically closed field is determined by wild ramification of the restrictions to all curves. We similarly deduce from it that so is the alternating sum of the Swan conductors of the cohomology groups, for a constructible sheaf on a variety over a local field.