Abstract
This thesis studies how the motives of hyper-Kähler varieties are controlled by smaller, "surface-like" motives. More precisely, we conjecture that the André motive of a hyper-Kähler variety is controlled by its component in degree two, which looks like the motive of a K3 surface. To tackle this problem we attach to any hyper-Kähler variety its defect group, an algebraic group which measures the failure of our prediction and enjoys several nice properties. We confirm the conjecture for all hyper-Kähler varieties of known deformation type. As a consequence we prove that all Hodge classes
on known hyper-Kähler varieties are motivated, and that their motives are abelian and satisfy the Mumford-Tate conjecture. IIn general, for potentially yet to be discovered examples, we prove that deformation equivalent hyper-Kähler varieties with Hodge-isometric second cohomology have isomorphic André motives.
