Arithmetic and Geometry pp 361-394 | Cite as

# A Crystalline Torelli Theorem for Supersingular K3 Surfaces

## Abstract

I like to argue that crystalline cohomology will play a role in characteristic *p* analogous to the role of Ilodge theory in characteristic zero. One aspect of this analogy is that the *F*-crystal structure on crystalline cohomology should reflect deep geometric properties of varieties. This should be especially true of varieties for which the “*p*-adic part” of their geometry is the most interesting, as seems, often to be true of supersingular varieties in the sense of Shioda [26]. For example, in [19 §6] I proved that supersingular abelian varieties of dimension at least two are determined up to isomorphism by the *F*-crystal structure (and trace map) on H _{ cris } ^{1} , just as abelian varieties over ℂ are determined by the Ilodge structure on H _{ DR } ^{1} .

## Keywords

Spectral Sequence Finite Type Ample Line Bundle Algebraic Space Kummer Surface## Preview

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