  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.
