Beauville–Voisin conjecture for all projective irreducible holomorphic symplectic varieties

Prove that, for every projective irreducible holomorphic symplectic variety X, the cycle class map is injective on the rational Chow subalgebra generated by divisor classes and the Chern classes of the tangent bundle.

Background

The Beauville–Voisin conjecture generalizes the Beauville–Voisin description of distinguished cycles on K3 surfaces to higher-dimensional projective irreducible holomorphic symplectic varieties. It asserts that algebraic relations among divisor classes and tangent-bundle Chern classes in the Chow ring are completely detected by their cohomology classes.

The paper proves this conjecture for smooth Debarre–Voisin fourfolds of Picard number one, and therefore for very general Debarre–Voisin fourfolds. The conjecture remains stated in general for arbitrary projective irreducible holomorphic symplectic varieties, beyond the cases known in the paper.

References

Beauville conjectured that the cycle class map is injective on the subring generated by divisor classes. Voisin extended this conjecture to include the Chern classes of the tangent bundle Conjecture 1.3. Let X be a projective irreducible holomorphic symplectic variety, and let

R*(X)\colonequals \langle\CH1(X),c_i(T_X)\rangle_{Q} \subset\CH*(X)

be the Q-subalgebra generated by divisor classes and the Chern classes of the tangent bundle. Then \cl|_{R*(X)} is injective.

— Ambient Chow Rings of Debarre--Voisin Fourfolds and the Beauville--Voisin Conjecture  (2610.23376 - Akazawa, 16 Oct 2026) in Section 1, Introduction, Conjecture 1.3 (Conjecture Beauville–Voisin)