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

Abstract

Let $C=A(r,r^{\prime })$ be a closed annulus of radii $r$ and $r^{\prime }$ ($r<r^{\prime }\in {\mathbb{Q}}_{\ge 0}$) over a complete discrete valuation field with algebraically closed residue field of characteristic $p>0$. To an étale 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 ${\mathrm{sw}}_{\mathrm{AS}}(\mathcal{F},\cdot ):[r,r^{\prime }]\cap {\mathbb{Q}}_{\ge 0}\to \mathbb{Q}$ which, for the variable $t$, measures the ramification of $\mathcal{F}|C^{[t]}$ - 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 $t^{\prime }$ 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 $t^{\prime }$ as the difference of the orders of the characteristic cycles of $\mathcal{F}$ at $t$ and $t^{\prime }$.

34 pages. Comments are welcome