Séminaire de Géométrie Arithmétique Paris-Pékin-Tokyo

# Deformation and rigidity of $\ell$-adic sheaves

## by Prof. Lei FU (Tsinghua University)

Europe/Paris
Centre de conférences Marilyn et James Simons (IHES)

### Centre de conférences Marilyn et James Simons

#### IHES

Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette
Description

Let X be a smooth connected algebraic curve over an algebraically closed field, let S be a finite closed subset in X, and let F_0 be a lisse l-torsion sheaf on X-S. We study the deformation of F_0. 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 $\overline{Q}_\ell$-sheaf F is irreducible and physically rigid, then it is cohomologically rigid in the sense that \chi(X,j_*End(F))=2, where j:X-S--> X is the open immersion.

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