On the motive of some hyper-Kaehler varieties of O'Grady-10 type.
Lie Fu (Univ. Claude Bernard Lyon 1)

The general question is to investigate how the motive of a hyper-Kaehler variety can be obtained by tensor operations from a surface-like (weight-2) motive. I will present two ways to make this precise, mainly focus in the case of O'Grady-10 type varieties. First, we show that the Chow motive of O'Grady-10-type crepant resolutions of moduli space of semistable sheaves on a K3 surface is in the tensor subcategory generated by the Chow motive of the surface. As consequences, we show the standard conjecture for those resolutions and we obtain results on the Voevodsky motive of the (open) moduli space of stable locus and the original singular moduli space. Second, we show that the André motive of any hyper-Kähler variety of O'Grady-10 deformation type lies in the tensor subcategory generated by its degree-2 part. As a consequence, their motive is of abelian type and the Mumford-Tate conjecture holds for them. This is a joint work with Salvatore Floccari and Ziyu Zhang.