Algebraic cycles and EPW cubes

Abstract

Let $X$ be a hyperk\"ahler variety with an anti-symplectic involution $\iota$. According to Beauville's conjectural "splitting property", the Chow groups of $X$ should split in a finite number of pieces such that the Chow ring has a bigrading. The Bloch-Beilinson conjectures predict how $\iota$ should act on certain of these pieces of the Chow groups. We verify part of this conjecture for a $19$-dimensional family of hyperk\"ahler sixfolds that are "double EPW cubes" (in the sense of Iliev-Kapustka-Kapustka-Ranestad). This has interesting consequences for the Chow ring of the quotient $X/\iota$, which is an "EPW cube" (in the sense of Iliev-Kapustka-Kapustka-Ranestad).Comment: 32 pages, to appear in Math. Nachrichten, feedback welcom

On Chow groups of some hyperkahler fourfolds with a non-symplectic involution, II

“…This conjecture is studied (and proven in some favourable cases) in [14], [15], [16], [17], [18]. The aim of this article is to provide more examples where conjecture 1.1 is verified, by considering Fano varieties of lines on cubic fourfolds.…”

On the Chow Ring of Certain Lehn–lehn–sorger–van Straten Eightfolds

“…for any ample divisor class h ∈ A 1 (Z) (see [34,Corollary 3.2]). Taking a g-invariant ample class h, this gives in particular an isomorphism…”

Algebraic cycles and Lehn–Lehn–Sorger–van Straten eightfolds

“…Thus, it will suffice to prove that for a big -invariant divisor on , there are isomorphisms The case is proven in [31, Theorem 3.1]; the general case is only notationally more complicated.…”