Claire Voisin Jussieu Title: Abel-Jacobi map, integral Hodge classes and decomposition of the diagonalAbstract: Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$,parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this talk 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. |