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

## TL;DR

This paper proves that the property of having an abelian André motive is preserved under deformation for projective hyperkähler manifolds, leading to new results on Hodge classes and the Mumford-Tate conjecture.

## Contribution

It establishes the invariance of the André motive's abelian property under deformation and applies this to key hyperkähler deformation types.

## Key findings

- André motives are abelian for K3^[n], generalized Kummer, and OG6 types.
- All Hodge classes are absolute on these manifolds.
- The Mumford-Tate conjecture holds for even degree cohomology of such manifolds.

## 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.