Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal

By Claire Voisin. 2010

54 citations recorded by the source corpus.

Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of 1-cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal. We also study this property for cubic threefolds, completing the work of Iliev-Markoushevich. We then conclude that the Hodge conjecture holds for degree 4 integral Hodge classes on fibrations into cubic threefolds over curves, with restriction on singular fibers.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/41657aa5055d167a4023b636af501cada2460801

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (2014, methodology): “It is also solved affirmatively in [113] for cubic threefolds.”