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Galois representations on the cohomology of hyper-Kähler varieties

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 $f\colon H^2(X_1,\mathbb{Q})\to H^2(X_2,\mathbb{Q})$, 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.