On the birational motive of hyper-Kähler varieties (Q2145845)

The author introduces in a motivic setting, an ascending filtration on the Chow groups CH\(_i(X)\) on a smooth projective variety \(X\) equipped with a unit \(o \in \) CH\(_0(X)\), called the co-radical filtration. Splitting of filtrations on the Chow group of zero-cycles of hyper-Kähler varieties has a motivation from the existence of Bloch-Beilinson filtrations, see [\textit{A. Beauville}, Lond. Math. Soc. Lect. Note Ser. 344, 38--53 (2007; Zbl 1130.14006)]. In the case when the variety \(X\) is birational to one of the following hyper-Kähler varieties of dimension \(2n\): a moduli space of stable sheaves on a \(K3\) surface, the generalized Kummer variety of an abelian surface, or the Fano variety of lines on a smooth cubic fourfold, the author shows that the \(k\)-factor of the co-radical filtration on CH\(_0(X)\) coincides with the group spanned by classes of points supported on a closed subvariety of dimension \(\geq n-k\) such that their points are rationally equivalent to \(X\) (i.e, the \(k\)-factor in the filtration introduced by \textit{C. Voisin} [Prog. Math. 315, 365--399 (2016; Zbl 1352.32010)]. Moreover, just in the case when \(X\) is birational to a moduli space of stable sheaves on a \(K3\) surface, the co-radical filtration corresponds to the filtration in [\textit{J. Shen} et al., Compos. Math. 156, No. 1, 179--197 (2020; Zbl 1436.14071)]. Under the assumption that \(X\) is one of the previous cases, the author proves that the birational Chow motive \(\mathfrak{h}^0(X)\) with rational coefficients admits a co-multiplicative birational Chow-Künneth decomposition that enriches to \(\mathfrak{h}^0(X)\) of a co-algebra grading structure. This new approach induces a natural description in the birational motive \(\mathfrak{h}^0(X)\) of such hyper-Kähler varieties (including also when \(X\) is the LLSvS eightfold) as the birational motive on the surface decomposition of \(X\), as in the sense of [\textit{C. Voisin}, ``Triangle varieties and surface decomposition of hyper-Kähler manifolds'', Preprint, \url{arXiv:1810.11848}].