Tohoku Mathematical Journal
2025
December
SECOND SERIES VOL. 77, NO. 4

Tohoku Math. J.
77 (2025), 499-537

Title VARIATION OF THE SWAN CONDUCTOR OF AN $\MATHBB{F}_\ELL$-SHEAF ON A RIGID ANNULUS

Author Amadou Bah

(Received November 21, 2022, revised January 15, 2024)
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 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 $\mathrm{sw}(\mathcal{F}, \cdot): [r, r'] \to \mathbb{Q}$ which, for the variable $t$, measures the ramification of the restriction of $\mathcal{F}$ to the sub-annulus of $C$ of radius $t$ with 0-thickness, along the special fiber of the normalized integral model of said sub-annulus. 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 $\mathrm{sw}(\mathcal{F}, \cdot)$ at $t$ and $t'$ as the difference of the orders of the characteristic cycles of $\mathcal{F}$ at $t$ and $t'$.

Mathematics Subject Classification. Primary 14F20; Secondary 14G22, 11S15.
Key words and phrases. Abbes-Saito ramification theory, characteristic cycle, rigid annulus, Swan conductor, nearby cycles.

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