On the Chow ring of double EPW quartics

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.