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

TL;DR

This paper studies the variation of the Swan conductor function for tale sheaves on a rigid annulus, proving its continuity, convexity, piecewise linearity, and relating slope differences to characteristic cycle orders.

Contribution

It introduces a detailed analysis of the Swan conductor function's behavior on rigid annuli, establishing its properties and linking slope changes to characteristic cycle variations.

Findings

The Swan conductor function is continuous, convex, and piecewise linear.

Slopes of the Swan conductor are integers.

Differences in slopes correspond to differences in characteristic cycle orders.

Abstract

Let $C=A(r, r')$ be a closed annulus of radii $r$ and $r'$ ($r < r' \in \mathbb{Q}_{\geq 0}$) over a complete discrete valuation field with algebraically closed residue field of characteristic $p>0$. To an \'etale sheaf of $\mathbb{F}_{\ell}$-modules $\mathcal{F}$ on $C$, ramified at most at a finite set of rigid points of $C$, we associate an Abbes-Saito Swan conductor function $\mathrm{sw}_{\mathrm{AS}}(\mathcal{F}, \cdot): [r, r']\cap \mathbb{Q}_{\geq 0} \to \mathbb{Q}$ which, for the variable $t$, measures the ramification of $\mathcal{F}\lvert C^{[t]}$ - the restriction of $\mathcal{F}$ to the sub-annulus $C^{[t]}$ of $C$ of radius $t$ with $0$-thickness - along the special fiber of the normalized integral model of $C^{[t]}$. We show that this function is continuous, convex and piecewise linear outside the radii of the ramification points of $\mathcal{F}$, with finitely many slopes which are all integers. For two distinct radii $t$ and $t'$ lying between consecutive radii of ramification points of $\mathcal{F}$, we compute the difference of the slopes of $\mathrm{sw}_{\mathrm{AS}}(\mathcal{F}, \cdot)$ at $t$ and $t'$ as the difference of the orders of the characteristic cycles of $\mathcal{F}$ at $t$ and $t'$.