l'Institut des Hautes Études Scientifiques, le Morningside Center of Mathematics, Chinese Academy of Sciences
et le Département des Sciences Mathématiques de l'Université de Tokyo

Ahmed Abbes (CNRS, IHÉS), Fabrice Orgogozo (CNRS, École polytechnique), Takeshi Saito (Université de Tokyo),
Atsushi Shiho (Université de Tokyo), Ye Tian (Morningside Center for Mathematics), Yichao Tian (Morningside Center for Mathematics),
Takeshi Tsuji (Université de Tokyo), Weizhe Zheng (Morningside Center for Mathematics)

Mercredi 21 mai 2014 de 10h30 à 11h30

Shenghao Sun (Mathematical Sciences Center of Tsinghua University)

Parity of Betti numbers in étale cohomology

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.