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arXiv · 1904.11320

Deformation principle and Andr\'e motives of projective hyperk\"ahler manifolds

Andrey Soldatenkov
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Abstract

Let $X_1$ and $X_2$ be deformation equivalent projective hyperk\"ahler manifolds. We prove that the Andr\'e motive of $X_1$ is abelian if and only if the Andr\'e motive of $X_2$ is abelian. Applying this to manifolds of $\mbox{K3}^{[n]}$, generalized Kummer and OG6 deformation types, we deduce that their Andr\'e motives are abelian. As a consequence, we prove that all Hodge classes in arbitrary degree on such manifolds are absolute. We discuss applications to the Mumford-Tate conjecture, showing in particular that it holds for even degree cohomology of such manifolds.

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Andrey Soldatenkov. 2019-04-25. Deformation principle and Andr\'e motives of projective hyperk\"ahler manifolds. https://arxiv.org/abs/1904.11320

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