Nearby slopes and boundedness for $\ell$-adic sheaves in positive characteristic

TL;DR

This paper investigates the boundedness of nearby slopes of -adic sheaves in positive characteristic, proving finiteness on curves and vanishing in specific cases, advancing understanding of sheaf ramification.

Contribution

It introduces a boundedness conjecture for nearby slopes and proves it for smooth curves, including finiteness and vanishing results for specific sheaves.

Findings

Finiteness of nearby slopes for constructible -adic sheaves.

Vanishing of nearby slopes for constant sheaves under semi-stable reduction.

Motivates a broader boundedness conjecture in positive characteristic.

Abstract

The goal of this paper is to motivate a boundedness conjecture on nearby slopes of $\ell$-adic sheaves in positive characteristic, and to prove it for smooth curves. For a constructible $\ell$-adic sheaf, we prove the finiteness of the set of nearby slopes associated to a given morphism. For the constant sheaf, we prove the vanishing of nearby slopes in case of generalized semi-stable reduction.