Ramification and nearby cycles for l-adic sheaves on relative curves

TL;DR

This paper revisits a formula for the dimension of nearby cycles of l-adic sheaves on relative curves, using Abbes-Saito's ramification theory to provide a new proof.

Contribution

It offers a new proof of Deligne and Kato's formula by applying Abbes-Saito's ramification theory, enhancing understanding of nearby cycles in algebraic geometry.

Findings

Reproves the dimension formula for nearby cycles using ramification theory

Connects ramification theory with the computation of nearby cycles

Provides a new perspective on the behavior of l-adic sheaves on relative curves

Abstract

Deligne and Kato proved a formula computing the dimension of the nearby cycles complex of an l-adic sheaf on a relative curve over an excellent strictly henselian trait. In this article, we reprove this formula using Abbes-Saito's ramification theory.