Curve classes on irreducible holomorphic symplectic varieties

Abstract

We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3 type and of Generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.Corollary 0.3. The integral Hodge conjecture holds for 2-cycles on cubic fourfolds.The proofs in this paper rely on several results and constructions that were already in the literature. In particular, Theorems 0.1 and 0.2 involve a deformation argument similar to that in [AV] and [CP]. We first consider the Hilbert scheme of a K3 or a generalized Kummer variety, where we exhibit special families of rational curves that represent primitive classes in H 2n−2 (X, Z), and which also deform in their Hodge loci. This then in turn implies that any integral degree 2n − 2 Hodge class on a deformation is represented by a rational curve.We would like to thank O. Benoist, G. Pacienza, M. Shen and C.

Rationality, universal generation and the integral Hodge conjecture

“…If we assume that the integral Hodge conjecture holds for 1-cycles on F , then the term Σ can always be absorbed into θ. Note that this assumption has recently been established by Mongardi-Ottem [17]. Proposition 5.3.…”

“…is decomposable. Since the integral Hodge conjecture holds for X (see Voisin [28]), we know that (17) Γ…”

“…for some γ ∈ CH 1 (X) and some symmetric cycle Σ ∈ CH 2 (X) ⊗ CH 2 (X). Since CH 2 (F ) → CH 1 (X) is surjective and there exists τ ∈ CH 2 (F ) such that P * τ = h, the term γ ⊗ h + h ⊗ γ can be absorbed into the term (P × P ) * θ and hence we can assume that γ = 0 in equation (17). There also exists τ ′ ∈ CH 1 (F ) such that P * τ ′ = h 2 in cohomology.…”

“…The integral Hodge conjecture for 1-cycles on F has recently been established by Mongardi-Ottem[17].…”

Universal generation of the cylinder homomorphism of cubic hypersurfaces

“…As a result of the surjectivity of the cylinder homomorphism, we prove the integral Hodge conjecture for 1-cycles on F (X) in Theorem 3.1, namely any integral Hodge class in H 6 (F (X), Z) is algebraic. Recently, G. Mongardi and J. C. Ottem [10] proved the integral Hodge conjecture for 1-cycles on hyper-Kähler varieties of K3-type and the generalized Kummer type. Our conclusion provides a different proof of a particular case of them.…”