Relative singular support and the semi-continuity of characteristic cycles for \'etale sheaves

TL;DR

This paper extends the concept of singular support to a relative setting for étale sheaves, proving generic constancy of characteristic cycles in fibrations and highlighting differences in semi-continuity behavior in higher dimensions.

Contribution

It introduces a relative notion of singular support for étale sheaves and demonstrates its properties, including generic constancy and semi-continuity issues in higher dimensions.

Findings

Proves generic constancy of singular supports and characteristic cycles in smooth fibrations.

Shows failure of lower semi-continuity of characteristic cycles in higher relative dimensions.

Highlights differences from known results in the relative curve case.

Abstract

Recently, the singular support and the characteristic cycle of an \'etale sheaf on a smooth variety over a perfect field are constructed by Beilinson and Saito, respectively. In this article, we extend the singular support to a relative situation. As an application, we prove the generic constancy for singular supports and characteristic cycles of \'etale sheaves on a smooth fibration. Meanwhile, we show the failure of the lower semi-continuity of characteristic cycles in a higher relative dimension case, which is different from Deligne and Laumon's result in the relative curve case.