Algebraic cycles and very special cubic fourfolds

Abstract

This note is about the Chow ring of Verra fourfolds. For a general Verra fourfold, we show that the Chow group of homologically trivial 1-cycles is generated by conics. We also show that Verra fourfolds admit a multiplicative Chow-Künneth decomposition, and draw some consequences for the intersection product in the Chow ring of Verra fourfolds. 1 2 ROBERT LATERVEERCorollary (=Corollary 5.3). Let X be any Verra fourfold. The subalgebrainjects into cohomology via the cycle class map. (Here, c j (T X ) are the Chow-theoretic Chern classes.) (This can be generalized to self-products X m , cf. Corollary 5.6.) The construction of a multiplicative Chow-Künneth decomposition relies once more on the spread argument, by first showing (Theorem 4.8) that any Verra fourfold can be motivically related to a genus 2 K3 surface. This close link to K3 surfaces also has the following consequence:Corollary (=Corollary 5.9). Let X, X ′ be two Verra fourfolds that are isogenous (i.e., there exists an isomorphism of Q-vector spaces H 4 (X, Q) ∼ = H 4 (X ′ , Q) compatible with the Hodge structures and with cup product). Then there is an isomorphism of Chow motivesFurther consequences concern a multiplicative decomposition in the derived category (Corollary 5.12), the generalized Hodge conjecture for self-products, and finite-dimensionality of the motive of certain special Verra fourfolds (Corollary 5.14).Conventions. In this note, the word variety will refer to a reduced irreducible scheme of finite type over C. For any n-dimensional variety X, we will write A i (X) for the Chow group of dimension i cycles on X with Q-coefficients, and A j (X) for the operational Chow cohomology of [17] with Q-coefficients (this can be identified with A n−j (X) for smooth X).The notations A j hom (X) and A j AJ (X) will indicate the subgroups of homologically trivial (resp. Abel-Jacobi trivial) cycles.For a morphism between smooth varieties f : X → Y , we will write Γ f ∈ A * (X × Y ) for the graph of f , and t Γ f ∈ A * (Y × X) for the transpose correspondence.The contravariant category of Chow motives (i.e., pure motives with respect to rational equivalence as in [43], [39]) will be denoted M rat .We will write H * (X) = H * (X, Q) for singular cohomology with Q-coefficients. VERRA FOURFOLDSDefinition 2.1 ([50]). A Verra fourfold is a double cover of P 2 × P 2 branched along a smooth divisor of bidegree (2, 2). Remark 2.2. The moduli space of Verra fourfolds is 19-dimensional [25, Section 0.3]. Lemma 2.3. A Verra fourfold X is a Fano variety (i.e., the canonical bundle is anti-ample). In particular, A 4 (X) = Q and the Hodge conjecture is true for X. Proof. The canonical bundle formula shows that X is Fano. Fano varieties are rationally connected [13], [30], and so A 4 (X) = Q. The Hodge conjecture is known for fourfolds with trivial Chow group of 0-cycles [10].