Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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


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: Given a smooth projective $ n$-fold $ Y$, with $ H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing codimension $ 2$-cycles in $ Y$ to the intermediate Jacobian $ J(Y)$, which is an abelian variety. Assuming $ n=3$, 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 with trivial Brauer group, we relate this property to the existence of an integral cohomological decomposition of the diagonal of $ Y$. 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 $ 4$ integral Hodge classes on fibrations into cubic threefolds over curves, with some restriction on singular fibers.


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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

American Mathematical Society