Let X be a holomorphic symplectic variety, let operatornameCH(X)_(0) denote the degree-zero part of the Chow grading associated with a Chow--Künneth decomposition, and let c_i(X) be its Chern classes. Multiplicative self-dual Chow--Künneth conjecture. The variety X admits a multiplicative self-dual Chow--Künneth decomposition such that c_i(X)∈ operatornameCH(X)_(0) for every i. This is a motivic reformulation of Beauville's splitting principle and is known in several cases, including K3 surfaces, Hilbert schemes of points on K3 or abelian surfaces, and generalized Kummer varieties.

The multiplicative self-dual Chow--Künneth conjecture for holomorphic symplectic varieties

The conjecture asks whether every smooth projective holomorphic symplectic variety has a multiplicative, self-dual Chow--Künneth decomposition with every Chern class $c_i(X)$ in its degree-zero Chow piece. The general existence question remains unresolved.

Known results

K3 surfaces and their Hilbert schemes admit decompositions of the conjectural type, with Chern classes in $\operatorname{CH}(X)_{(0)}$ .
Generalized Kummer varieties admit multiplicative Chow--Künneth decompositions (Fu--Tian--Vial).
The same framework covers Hilbert schemes of points on abelian surfaces and products or varieties birational to the established examples.
Matsushita’s results for relevant Lagrangian fibrations imply Beauville’s weak splitting property in several deformation types, but not the general conjecture.

Current status (as of August 2026): The conjecture is established for the recorded K3, Hilbert-scheme, generalized-Kummer, and related birational cases, but no general proof or counterexample has been publicly reported.

The multiplicative self-dual Chow--Künneth conjecture for holomorphic symplectic varieties

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