On the Mumford-Tate conjecture for hyperkähler varieties

Abstract: We study the Mumford--Tate conjecture for hyperk\"{a}hler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O'Grady's ten dimensional example, and all of their self-products. For an arbitrary hyperk\"{a}hler variety whose second Betti number is not 3, we prove the Mumford--Tate conjecture in every codimension under the assumption that the K\"{u}nneth components in even degree of its Andr\'{e} motive are abelian. Our results extend a theorem of Andr\'{e}.