Symmetry and parity in Frobenius action on cohomology

Junecue Suh

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arXiv · 2410.01184

Symmetry and parity in Frobenius action on cohomology

Junecue Suh

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Abstract

We prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincar\'e duality and the hard Lefschetz theorem. As a corollary, we deduce that the Betti numbers in odd degrees of any proper smooth variety over a field are even (a consequence of Hodge symmetry in characteristic zero), answering an old question of Serre. Then we give a generalization and a refinement for arbitrary varieties over finite fields, in response to later questions of Serre and of Katz.

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BibTeX RIS

Junecue Suh. 2024-10-02. Symmetry and parity in Frobenius action on cohomology. https://arxiv.org/abs/2410.01184

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