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Symmetry and parity in Frobenius action on cohomology

Published online by Cambridge University Press:  08 December 2011

Junecue Suh*
Affiliation:
Department of Mathematics, FAS Harvard University, One Oxford Street, Cambridge, MA 02138, USA (email: jsuh@math.harvard.edu)
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Abstract

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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 Poincaré 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.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2011

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