On the motive of O'Grady's six dimensional hyper-K\"{a}hler varieties

TL;DR

This paper proves that certain six-dimensional hyper-K"{a}hler varieties related to O'Grady's construction have Chow motives generated by the motive of an abelian surface, confirming key conjectures in specific cases.

Contribution

It provides a formula for the Chow motive of OG6-type hyper-K"{a}hler varieties in terms of the surface's motive, establishing new links between their motives.

Findings

Chow motives of OG6-type varieties are generated by the motive of the abelian surface.

The paper confirms the Hodge and Tate conjectures for many OG6-type hyper-K"{a}hler varieties.

A formula for the Chow motive of these varieties in terms of the surface's motive is provided.

Abstract

We prove that the rational Chow motive of a six dimensional hyper-K\"{a}hler variety obtained as symplectic resolution of O'Grady type of a singular moduli space of semistable sheaves on an abelian surface $A$ belongs to the tensor category of motives generated by the motive of $A$. We in fact give a formula for the rational Chow motive of such a variety in terms of that of the surface. As a consequence, the conjectures of Hodge and Tate hold for many hyper-K\"{a}hler varieties of OG6-type.