Séminaire de Géométrie Arithmétique Paris-Pékin-Tokyo

# Parity of Betti numbers in étale cohomology

## by Prof. Shenghao SUN (Mathematical Sciences Center of 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
By Hodge symmetry, the Betti numbers of a complex projective smooth variety in odd degrees are even. When the base field has characteristic p, Deligne proved the hard Lefschetz theorem in étale cohomology, and the parity result follows from this. Suh has generalized this to proper smooth varieties in characteristic p, using crystalline cohomology. The purity of intersection cohomology group of proper varieties suggests that the same parity property should hold for these groups in characteristic p. We proved this by investigating the symmetry in the categorical level. In particular, we reproved Suh's result, using merely étale cohomology. Some related results will be discussed. This is joint work with Weizhe Zheng.

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