Double EPW sextics and the Voisin filtration on zero-cycles
Michele Bolognesi, Robert Laterveer
TL;DR
This work establishes a precise link between the $ι$-anti-invariant part of zero-cycles on a double EPW sextic $X$ and Voisin’s rational orbit filtration by proving $A^4(X)^- = S_1A^4(X)\cap A^4_{hom}(X)$. It also shows that, for a very general $X$, the anti-invariant part of its Chow motive decomposes in terms of the motive of a Gushel–Mukai fourfold $Y$, yielding a motivic isomorphism and a correspondence $\Gamma$ between $A^3_{hom}(Y)$ and $S_1A^4(X)$. The results rely on the Iliev–Manivel, modular, and Franchetta frameworks, quadratic/linear relations among natural correspondences, and Voisin’s filtration, plus a spread-argument to transfer results across families. Consequences include Chow–Lefschetz isomorphisms for $X$, a Chow–Abel–Jacobi isomorphism linking $Y$ and $X$, and the verification of the conjecture for Fano varieties of lines of very general cubic fourfolds in the Hassett divisor $\mathcal{C}_{12}$, where infinite-order birational automorphisms occur. Altogether, the paper advances the understanding of how anti-symplectic symmetries control zero-cycles and their motives on double EPW sextics and related hyper-Kähler varieties.
Abstract
Let $X$ be a double EPW sextic, and $ι$ its anti-symplectic involution. We relate the $ι$-anti-invariant part of the Chow group of zero-cycles of $X$ with Voisin's rational orbit filtration. For a general double EPW sextic $X$, we also relate the anti-invariant part of the Chow motive of $X$ with the motive of a Gushel-Mukai fourfold. As an application, we obtain a similar result for certain Fano varieties of lines in cubics with infinite-order birational automorphisms.