Deformations and Rigidity of ℓ-adic Sheaves

2016/11/12 by Fu, Lei
#14D15 #14G22 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1611.03964

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 \mathbb F_ℓ-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 \mathbb Q_ℓ-sheaf \mathcal F on X-S is irreducible and rigid, then we have dim H¹(X,j_∗\mathcal End(\mathcal F))=2g, where j:X-S→ X is the open immersion, and g is the genus of X.

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