Galois representations on the cohomology of hyper-Kähler varieties

Abstract: We show that the Andr\'{e} motive of a hyper-K\"{a}hler 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-K\"{a}hler varieties with $b_2(X_i)>6$ and if there exists a Hodge isometry $f\colon H^2(X_1,\mathbb{Q})\to H^2(X_2,\mathbb{Q})$, then the Andr\'e 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 \'{e}tale 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-K\"{a}hler varieties for which the Mumford--Tate conjecture is true.