# On the Mumford–Tate conjecture for hyperkähler varieties

@article{Floccari2019OnTM, title={On the Mumford–Tate conjecture for hyperk{\"a}hler varieties}, author={Salvatore Floccari}, journal={arXiv: Algebraic Geometry}, year={2019} }

We study the Mumford-Tate conjecture for hyperkahler varieties. Building on work of Markman, we show that it holds in arbitrary codimension for all varieties of $\mathrm{K}3^{[m]}$-type. For an arbitrary hyperkahler variety satisfying $b_2(X)>3$ we establish one of the two inclusions of algebraic groups predicted by the Mumford-Tate conjecture. Our results extend a theorem of Andre.

#### 2 Citations

Deformation Principle and André motives of Projective Hyperkähler Manifolds

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Let $X_1$ and $X_2$ be deformation equivalent projective hyperkähler manifolds. We prove that the André motive of $X_1$ is abelian if and only if the André motive of $X_2$ is abelian. Applying this… Expand

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We investigate how the motive of hyper-K\"ahler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular… Expand

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