THE MOTIVE OF THE HILBERT CUBE

MINGMIN SHEN,CHARLES VIAL

doi:10.1017/fms.2016.25

MINGMIN SHEN, CHARLES VIAL

Open Access

https://doi.org/10.1017/fms.2016.25

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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–Künneth decomposition is stable under taking the Hilbert cube. This is achieved by considering an explicit resolution of the rational map$X^{3}{\dashrightarrow}X^{[3]}$. The case of the Hilbert square was taken care of in Shen and Vial [Mem. Amer. Math. Soc.240(1139) (2016), vii+163 pp]. The archetypical examples of varieties endowed with a multiplicative Chow–Künneth decomposition is given by abelian varieties. Recent work seems to suggest that hyperKähler varieties share the same property. Roughly, if a smooth projective variety$X$has a multiplicative Chow–Künneth decomposition, then the Chow rings of its powers$X^{n}$have a filtration, which is the expected Bloch–Beilinson filtration, that is split.

Highlights

Let X be a smooth projective variety of dimension d, over a field k, which we assume is endowed with aChow–Kunneth decomposition {π i X ∈ CHd (X × X) :0 i 2d}
This means that the Kunneth decomposition of the -adic class ( = char k) of the diagonal [∆X ] ∈ H2d(Xk, Q ) is algebraic and lifts to a splitting of the rational Chow motive of CHd(X × X ) with rational coefficients such when seen as self-correspondences of X, π
In [12], we proved, under the technical, but yet natural, assumption that the Chern classes cp(X ) belong to the graded-0 part CHp(X )0 of CHp(X ), that if X is a smooth projective variety that admits a multiplicative Chow–Kunneth decomposition, the Hilbert scheme of length-2 subschemes X [2] admits a multiplicative

Summary

Introduction

Let X be a smooth projective variety of dimension d, over a field k, which we assume is endowed with a. In [12], we proved, under the technical, but yet natural, assumption that the Chern classes cp(X ) belong to the graded-0 part CHp(X )0 of CHp(X ), that if X is a smooth projective variety that admits a multiplicative Chow–Kunneth decomposition (for example X a K3 surface), the Hilbert scheme of length-2 subschemes X [2] admits a multiplicative. In Theorem 7.1, we establish the analogue of Theorem 1 in those cases, by showing that X [1,2] or X [2,3] admit a selfdual multiplicative Chow–Kunneth decomposition with Chern classes belonging to the graded-0 part of the Chow ring, whenever X has a self-dual multiplicative.

Self-dual Chow–Kunneth decompositions

Multiplicative Chow–Kunneth decompositions

Desingularizing the rational map X3

Findings

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