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

Abstract

Given a smooth projectivenn-foldYY, withH3,0(Y)=0H^{3,0}(Y)=0, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing codimension22-cycles inYYto the intermediate JacobianJ(Y)J(Y), which is an abelian variety. Assumingn=3n=3, we study in this paper the existence of families of11-cycles inYYfor which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. WhenYYitself is rationally connected with trivial Brauer group, we relate this property to the existence of an integral cohomological decomposition of the diagonal ofYY. We also study this property for cubic threefolds, completing the work of Iliev-Markushevich-Tikhomirov. We then conclude that the Hodge conjecture holds for degree44integral Hodge classes on fibrations into cubic threefolds over curves, with some restriction on singular fibers.