The motive of the Hilbert cube

TL;DR

This paper explores whether the property of having a multiplicative Chow-Kuenneth decomposition is preserved when constructing Hilbert cubes of smooth projective varieties, using explicit resolutions and previous results on Hilbert squares.

Contribution

It extends the understanding of Chow-Kuenneth decompositions to Hilbert cubes, building on prior work on Hilbert squares and considering explicit resolutions of related maps.

Findings

The property is stable under taking Hilbert cubes for certain varieties.

Explicit resolutions of the map to the Hilbert scheme are constructed.

The work relates to the structure of Chow rings and Bloch-Beilinson filtration.

Abstract

The Hilbert scheme $X^{[3]}$ of length-$3$ subschemes of a smooth projective variety $X$ is known to be smooth and projective. We investigate whether the property of having a multiplicative Chow-Kuenneth decomposition is stable under taking the Hilbert cube. This is achieved by considering an explicit resolution of the map $X^3 \dashrightarrow X^{[3]}$. The case of the Hilbert square was taken care of in previous work of ours. The archetypical examples of varieties endowed with a multiplicative Chow-Kuenneth decomposition is given by abelian varieties. Recent work seems to suggest that hyperKaehler varieties share the same property. Roughly, if a smooth projective variety $X$ has a multiplicative Chow-Kuenneth decomposition, then the Chow rings of its powers $X^n$ have a filtration, which is the expected Bloch-Beilinson filtration, that is split.