[Paper Review] The Chow ring of double EPW sextics

Andrea Ferretti|ArXiv.org|Jul 30, 2009

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TL;DR

This paper verifies a conjecture of Beauville and Voisin for very general double EPW sextics by proving that all polynomial relations between the hyperplane class h and the Chern classes of X that hold in cohomology also hold in the rational Chow group. Using degeneration techniques, cohomological computations, and Grothendieck-Riemann-Roch, the author establishes that in CH*(X)⊗Q, the relations h⁴ = 6θ, h²·c₂(X) = 60θ, c₂(X)² = 828θ, and c₄(X) = 324θ hold, with θ a class of a point on a specific subvariety, confirming the conjecture for this class of irreducible symplectic varieties.

ABSTRACT

A conjecture of Beauville and Voisin states that for an irreducible symplectic variety X, any polynomial relation between classes of divisors and the Chern classes of X which holds in cohomology already holds in the Chow groups. We verify the conjecture for a very general double EPW sextic.

Motivation & Objective

• To verify the Beauville-Voisin conjecture for double EPW sextics, which posits that cohomological relations between divisor and Chern classes hold in the rational Chow group.
• To establish that for a very general double EPW sextic X, all polynomial relations between h = f*O_Y(1) and Chern classes of X that hold in H*(X, Q) also hold in CH*(X)⊗Q.
• To compute the Chow ring relations involving h and Chern classes using degeneration, cohomological methods, and Grothendieck-Riemann-Roch.
• To demonstrate that the tautological relations in cohomology lift to the Chow group, particularly for the class θ corresponding to a point on Y_B[2].
• To extend the known cases of the conjecture beyond K3 surfaces and Fano varieties of lines in cubic fourfolds.

Proposed method

• Use of degeneration techniques to analyze the singular locus and deformation of double EPW sextics, particularly focusing on the behavior of the polarization h.
• Computation of Chern classes via Grothendieck-Riemann-Roch applied to the embedding of a Lagrangian subvariety Z ⊂ X, using the normal bundle and canonical bundle relations.
• Derivation of relations between h and Chern classes by comparing Chern classes of the tangent bundle and the bundle Q on the double cover X.
• Identification of the class θ as the pushforward of a point class from Y_B[2] ⊂ YA, and use of rational equivalence to relate h⁴ and θ in CH*(X)⊗Q.
• Application of the Lefschetz hyperplane theorem and cohomological computations to verify that the Chow relations match the cohomological ones.
• Use of the fact that CH₁(L₀) ≅ ℤ for rational curves L₀ to show that h⁴ is rationally equivalent to a multiple of θ, enabling precise coefficient determination via cohomological comparison.

Experimental results

Research questions

• RQ1 Does the Beauville-Voisin conjecture hold for double EPW sextics, i.e., do cohomological relations between divisor and Chern classes lift to the Chow group?
• RQ2 What are the precise Chow ring relations between the polarization h and the Chern classes c₂(X), c₄(X) in CH*(X)⊗Q for a double EPW sextic X?
• RQ3 How does the class θ, representing a point on Y_B[2], relate to h⁴ and other tautological classes in CH*(X)⊗Q?
• RQ4 Can degeneration techniques and Grothendieck-Riemann-Roch be used to compute Chern classes of bundles on double covers and verify their rational equivalence to cohomological relations?
• RQ5 Is the tautological relation h·c₂(X) = 5h³ in cohomology also valid in the Chow group for very general double EPW sextics?

Key findings

• The relation h⁴ = 6θ holds in CH₄(X)⊗Q, with θ the class of a point on Y_B[2], confirmed via rational equivalence on rational curves and cohomological comparison.
• The relation h²·c₂(X) = 60θ holds in CH₄(X)⊗Q, derived from pushforward of a tautological relation in CH₂(YA) and verified via cohomological computation.
• The square of the second Chern class satisfies c₂(X)² = 828θ in CH₈(X)⊗Q, established by squaring the pushforward of a tautological cycle.
• The fourth Chern class satisfies c₄(X) = 324θ in CH₈(X)⊗Q, confirmed by comparing with cohomological values after expressing c₄(Q) in terms of h and θ.
• The relation h·c₂(X) = 5h³ holds in CH₄(X)⊗Q, matching the cohomological relation and completing the verification of the conjecture.
• All tautological relations between h and Chern classes that hold in cohomology are shown to hold in CH*(X)⊗Q, confirming the Beauville-Voisin conjecture for very general double EPW sextics.