# On the Mumford-Tate conjecture for hyperk\"{a}hler varieties

## TL;DR

This paper proves the Mumford--Tate conjecture for certain hyperk"ahler varieties, including those deformation equivalent to specific known examples, and under certain motive conditions for others, extending previous results.

## Contribution

It establishes the Mumford--Tate conjecture for all varieties deformation equivalent to specific hyperk"ahler examples and under motive assumptions for others, broadening the conjecture's verified cases.

## Key findings

- Full conjecture holds for varieties deformation equivalent to Hilbert schemes or O'Grady's example.
- Proves the conjecture in every codimension for varieties with certain motive properties.
- Extends a theorem of Andre9 to new classes of hyperk"ahler 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}.