Semisimplicity, purity and Mumford-Tate conjecture for hyper-Kähler varieties

Kazuhiro Ito, Haitao Zou

We prove the Mumford-Tate conjecture in every degree for hyper-Kähler varieties over fields finitely generated over $\mathbb{Q}$. The proof establishes semisimplicity of $\ell$-adi…