Parity and symmetry in intersection and ordinary cohomology

Shenghao Sun, Weizhe Zheng

Published 2014-02-06, updated 2014-08-26Version 3

Suh showed recently that the odd-degree Betti numbers of proper smooth varieties are even, confirming a prediction of Deligne. In this paper, using a different approach, we show more generally that the odd-degree Betti numbers in intersection cohomology of proper varieties are even. We deduce this from a stability result of orthogonal and symplectic pure perverse sheaves under proper direct image. Over a finite field, the latter provides parity and symmetry results for Jordan blocks appearing in the Frobenius action on intersection cohomology groups. We show moreover that the subgroup of the Grothendieck group generated by orthogonal pure perverse sheaves of even weights and symplectic pure perverse sheaves of odd weights is stable under Grothendieck's six operations. In particular, we obtain virtual parity results for the Frobenius action on ordinary cohomology of nonproper varieties.