On the motive of some hyperKaehler varieties

TL;DR

This paper demonstrates that the motive of Hilbert schemes on K3 or abelian surfaces can be decomposed similarly to the motives of abelian varieties, extending known decompositions to new geometric contexts.

Contribution

It introduces a decomposition of motives for Hilbert schemes on K3 and abelian surfaces, paralleling classical decompositions for abelian varieties.

Findings

Motives of Hilbert schemes admit a similar decomposition to abelian varieties.

The decomposition aligns with known structures in algebraic geometry.

Extends the understanding of motives in hyperKähler geometry.

Abstract

We show that the motive of the Hilbert scheme of length-$n$ subschemes on a K3 surface or on an abelian surface admits a decomposition similar to the decomposition of the motive of an abelian variety obtained by Shermenev, Beauville, and Deninger and Murre.