On the Chow ring of double EPW quartics

TL;DR

This paper investigates the algebraic cycle structure of double EPW quartics, a special class of hyperk"ahler fourfolds, using their rich geometric properties and relations to moduli spaces and Verra fourfolds.

Contribution

It establishes general conjectures about algebraic cycles on hyperk"ahler varieties specifically for double EPW quartics, leveraging their unique geometric features.

Findings

Proves conjectures about algebraic cycles on double EPW quartics.

Links the geometry of double EPW quartics to moduli spaces of twisted sheaves.

Utilizes relations to Verra fourfolds to analyze algebraic cycles.

Abstract

Double EPW quartics are hyperk\"ahler varieties of dimension 4, first introduced by Iliev, Kapustka, Kapustka, and Ranestad. The general double EPW quartic is isomorphic to a moduli space of twisted sheaves on a $K3$ surface. They have a rich geometry: they are equipped with an anti-symplectic involution and are related to conics in Verra fourfolds in the same way Fano varieties of lines on cubic fourfolds are related to cubic fourfolds themselves. In this work, we exploit this geometry to establish general conjectures about algebraic cycles on hyperk\"ahler varieties in the case of double EPW quartics.