Galois representations on the cohomology of hyper-Kähler 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 $f:H^2(X_1,\mathbb{Q})\to H^2(X_2,\mathbb{Q})$, 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.

added Section 5; accepted for publication in Math. Zeit

References in corpus (3)

• Mapping class group and global Torelli theorem for hyperkahler manifolds: an erratum
• On the motive of O'Grady's ten-dimensional hyper-Kähler varieties
• On Irreducible Symplectic Varieties of ${K3}^{[n]}$-type in Positive Characteristic