THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$
By M. Shen, Charles Vial. Forum of Mathematics, Sigma, 2015
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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.
Subfields: Geometry and Topology, Mathematical Physics, Pharmacology
View Paper: https://www.semanticscholar.org/paper/4cf7b5033c39cff5d2577fb537ea24f7038c5ead
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Citations in the corpus
- The generalized Franchetta conjecture for some hyper-Kähler varieties, II (2020, background): “In both cases, the Chow motive of Y is known to be of Lefschetz type, so that it is finite-dimensional and any Chow–Künneth decomposition is multiplicative (see [SV16b, Th. 2]).”