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

Voisin, Claire

doi:10.1090/S1056-3911-2012-00597-9

Author: Claire Voisin
Journal: J. Algebraic Geom. 22 (2013), 141-174
DOI: https://doi.org/10.1090/S1056-3911-2012-00597-9
Published electronically: May 23, 2012
MathSciNet review: 2993050
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Abstract | References | Additional Information

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

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Additional Information

Claire Voisin
Affiliation: Institut de mathématiques de Jussieu, 175 rue due Chevaleret, 75013 Paris, France
Email: voisin@math.jussieu.fr

DOI: https://doi.org/10.1090/S1056-3911-2012-00597-9
Received by editor(s): April 17, 2010
Received by editor(s) in revised form: February 1, 2011
Published electronically: May 23, 2012
Dedicated: This paper is dedicated to the memory of Eckart Viehweg