Variation of the Swan conductor of an $\mathbb{F}_{\ell}$-sheaf on a rigid disc

TL;DR

This paper investigates how the Swan conductor of an $F_ ext{ extlambda}$-sheaf varies on a rigid disc, establishing its continuity, piecewise linearity, and relating slopes to characteristic cycles and ramification theory.

Contribution

It introduces a new function measuring ramification variation of sheaves on a rigid disc and characterizes its properties and slopes in terms of characteristic cycles.

Findings

The variation function is continuous and piecewise linear.

All slopes of the variation function are integers.

Slopes are computed via characteristic cycles and logarithmic differential forms.

Abstract

This article studies the variation of the Swan conductor of a lisse \'etale sheaf of $\mathbb{F}_{\ell}$-modules $\mathcal{F}$ on the rigid unit disc $D$ over a complete discrete valuation field $K$ with algebraically closed residue field of characteristic $p\neq \ell$. We associate to $\mathcal{F}$ a function ${\rm sw}_{\rm AS}(\mathcal{F}, \cdot): \mathbb{Q}_{\geq 0}\to \mathbb{Q}$, defined with the Abbes-Saito logarithmic ramification filtration, which measures, at each $t\in \mathbb{Q}_{\geq 0}$, the ramification of the restriction of $\mathcal{F}$ to the subdisc of radius $t$ along the special fiber of the normalized integral model. We prove that this function is continuous and piecewise linear, with finitely many slopes which are all integers. We compute the slope at $t\in \mathbb{Q}_{\geq 0}$ in terms of a characteristic cycle associated to $\mathcal{F}$, a (power of a) logarithmic differential form defined by ramification theory.