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The Beauville-Voisin conjecture for double EPW sextics

Robert Laterveer

TL;DR

This work proves the Beauville–Voisin conjecture for double EPW sextics, showing that the subalgebra generated by divisors and tangent Chern classes injects into cohomology. The authors extend the result to a strengthened statement incorporating $A^2(X)^+$ for $i\ge3$, and integrate a suite of techniques: Iliev–Manivel’s geometric construction, the modular PPZ viewpoint, the Franchettina property for generically defined cycles, quadratic and linear relations among incidence correspondences, and the Voisin–Bloch–Srinivas nilpotence argument. The argument leverages reductions via the Gushel–Mukai framework, spreaders across moduli, and a carefully crafted decomposition-of-the-diagonal to upgrade algebraic equivalence to rational equivalence. The methodology unites multiple perspectives on hyperkähler Chow rings and yields a robust injection result with implications for the motive and cycle theory of very general double EPW sextics.

Abstract

We prove that the Beauville-Voisin conjecture is true for any double EPW sextic, i.e. the subalgebra of the Chow ring generated by divisors and Chern classes of the tangent bundle injects into cohomology.

The Beauville-Voisin conjecture for double EPW sextics

TL;DR

This work proves the Beauville–Voisin conjecture for double EPW sextics, showing that the subalgebra generated by divisors and tangent Chern classes injects into cohomology. The authors extend the result to a strengthened statement incorporating for , and integrate a suite of techniques: Iliev–Manivel’s geometric construction, the modular PPZ viewpoint, the Franchettina property for generically defined cycles, quadratic and linear relations among incidence correspondences, and the Voisin–Bloch–Srinivas nilpotence argument. The argument leverages reductions via the Gushel–Mukai framework, spreaders across moduli, and a carefully crafted decomposition-of-the-diagonal to upgrade algebraic equivalence to rational equivalence. The methodology unites multiple perspectives on hyperkähler Chow rings and yields a robust injection result with implications for the motive and cycle theory of very general double EPW sextics.

Abstract

We prove that the Beauville-Voisin conjecture is true for any double EPW sextic, i.e. the subalgebra of the Chow ring generated by divisors and Chern classes of the tangent bundle injects into cohomology.
Paper Structure (11 sections, 22 theorems, 102 equations)

This paper contains 11 sections, 22 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Double EPW sextics: the original construction
  4. The Chow ring of double EPW sextics
  5. Double EPW sextics: the Iliev--Manivel construction
  6. Double EPW sextics: the modular construction
  7. Generically defined cycles and the Franchettina property
  8. A quadratic relation
  9. A linear relation
  10. Proof of the main result
  11. Erratum to "The Beauville--Voisin conjecture for double EPW sextics"

Key Result

Theorem

Let $X$ be a smooth double EPW sextic. The $\mathbb{Q}$-subalgebra injects into cohomology, under the cycle class map. In addition, let $A^2(X)^+\subset A^2(X)$ denote the subgroup of cycles invariant under the covering involution $\iota$. The cycle class map induces injections

Theorems & Definitions (55)

  • Conjecture 1.1
  • Theorem : =Theorem \ref{['main']}
  • Definition 2.1: Eisenbud--Popescu--Walter EPW
  • Theorem 2.2: O'Grady
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Definition 2.5
  • ...and 45 more