Semi-continuity for conductor divisors of \'etale sheaves

TL;DR

This paper establishes a semi-continuity property for conductor divisors of étale sheaves in higher dimensions, extending known results from curves to more complex geometric settings and drawing analogies with meromorphic connections.

Contribution

It generalizes semi-continuity results from relative curves to higher-dimensional cases for étale sheaves, providing a new higher-dimensional $\, ext{ extlbrackdbl}\, ext{ extgreater}$-adic analogy.

Findings

Proves semi-continuity for conductor divisors in higher dimensions

Extends semi-continuity from curves to higher relative dimensions

Provides an analogy with Poincaré-Katz ranks of meromorphic connections

Abstract

In this article, we prove a semi-continuity property for both conductor divisors and logarithmic conductor divisors for \'etale sheaves on higher relative dimensions in a geometric situation. It generalizes a semi-continuity result for conductors of \'etale sheaves on relative curves to higher relative dimensions, and it can be considered as a higher dimensional $\ell$-adic analogy of Andr\'e's result on the semi-continuity of Poincar\'e-Katz ranks of meromorphic connections on smooth relative curves.