Claire Voisin
Jussieu

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

Abstract: 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 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.
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