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On the Chow ring of double EPW quartics

Carl Mazzanti

Abstract

Double EPW quartics are hyperkähler 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 hyperkähler varieties in the case of double EPW quartics.

On the Chow ring of double EPW quartics

Abstract

Double EPW quartics are hyperkähler 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 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 hyperkähler varieties in the case of double EPW quartics.
Paper Structure (25 sections, 40 theorems, 180 equations)

This paper contains 25 sections, 40 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Algebraic cycles on hyperkähler varieties
  3. The Beauville--Voisin conjecture for double EPW quartics
  4. Parallels to Fano varieties of lines on cubic fourfolds
  5. Structure
  6. Conventions
  7. Acknowledgements
  8. The construction of double EPW quartics
  9. Double EPW quartics via conics in Verra fourfolds
  10. Double EPW quartics via Lagrangian degeneracy loci and their two Lagrangian fibrations
  11. Double EPW quartics as moduli spaces of twisted sheaves on K3 surfaces
  12. The Franchetta property
  13. The incidence variety
  14. The incidence variety in F(X)
  15. Pushing the incidence variety forward to Y
  16. ...and 10 more sections

Key Result

Theorem 1.3

Let $Y$ be a double EPW quartic. Then the Beauville--Voisin--Franchetta conjecture holds for $Y$, meaning that the cycle class map restricted to the subring generated by divisors and generically defined cycles, is injective.

Theorems & Definitions (92)

  • Conjecture 1.1: Beauville--Voisin, see \ref{['conj:BV']}
  • Conjecture 1.2: Franchetta conjecture, see \ref{['conj:franchetta']}
  • Theorem 1.3: \ref{['thm:BV']}
  • Theorem 1.4: \ref{['thm:constant-cycle']}
  • Theorem 1.5: \ref{['thm:chow-iso', 'prop:h^2', 'cor:chow^3_strong']}
  • Proposition 1.6: \ref{['prop:filtration']}
  • Theorem 1.7: \ref{['them:multiplicativity+0']}, \ref{['thm:A-multiplicativity']}
  • Theorem 2.1: ikkr17
  • Definition 3.1
  • Definition 3.2
  • ...and 82 more