Deformations and Rigidity of $\ell$-adic Sheaves

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 $\dim H^1(X,j_{\ast }\mathcal{E}nd(\mathcal{F}))=2g$, where $j:X-S\to X$ is the open immersion, and $g$ is the genus of $X$.

To appear in Advances in Mathematics

References in corpus (1)

• Revêtements étales et groupe fondamental (SGA 1)