Deformations and Rigidity of $\ell$-adic Sheaves

TL;DR

This paper investigates the deformation theory of $ar{F}_ ext{ell}$-sheaves on algebraic curves, proving a conjecture relating rigidity, irreducibility, and the dimension of a specific cohomology group using rigid analytic geometry.

Contribution

It establishes a link between the deformation space of $ar{F}_ ext{ell}$-sheaves and a conjecture of Katz, demonstrating a new geometric approach to understanding rigidity and cohomology dimensions.

Findings

Proves Katz's conjecture on the dimension of $H^1$ for rigid irreducible sheaves.

Shows the universal deformation space has a rigid analytic structure.

Connects deformation theory with classical algebraic geometry invariants.

Abstract

Let $X$ be a smooth connected projective algebraic curve over an algebraically closed field, and let $S$ be a finite nonempty closed subset in $X$. We study deformations of $\overline{\mathbb F}_\ell$-sheaves. The universal deformation space is a formal scheme. Its generic fiber has a rigid analytic space structure. By studying this rigid analytic space, we prove a conjecture of Katz which says that if a lisse $\overline{\mathbb Q}_\ell$-sheaf $\mathcal F$ on $X-S$ is irreducible and rigid, then we have $\mathrm{dim}\, H^1(X,j_\ast\mathcal End(\mathcal F))=2g$, where $j:X-S\to X$ is the open immersion, and $g$ is the genus of $X$.