Galois representations on the cohomology of hyper-K\xe4hler varieties

Abstract

We show that the André motive of a hyper-Kähler variety X over a field K subset {mathbb {C}} with b_2(X)>6 is governed by its component in degree 2. More precisely, we prove that if X_1 and X_2 are deformation equivalent hyper-Kähler varieties with b_2(X_i)>6 and if there exists a Hodge isometry f:H^2(X_1,{mathbb {Q}})rightarrow H^2(X_2,{mathbb {Q}}), then the André motives of X_1 and X_2 are isomorphic after a finite extension of K, up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the étale cohomology of X_1 and X_2 are isomorphic as well. We prove a similar result for varieties over a finite field which can be lifted to hyper-Kähler varieties for which the Mumford–Tate conjecture is true.