Invariants de Hasse-Witt des réductions de certaines variétés symplectiques irréductibles

Abstract

Let X be an irreducible symplectic variety defined over a number field K. Assume either that X has Picard number at least two or that X has even second Betti number. We prove that there exist a finite algebraic field extension L/K and a density 1 set S of non-archimedean places of L such that the reduction of X at any place in S has nonzero Hasse-Witt invariant.

arXiv comment15 pages, french. D. Huybrechts pointed out to me that Tate's conjecture was already known to be true for hyperkähler manifolds with second Betti number > 3 (a result due to Y. André)