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Hodge loci and absolute Hodge classes

Published online by Cambridge University Press:  17 July 2007

Claire Voisin
Claire Voisin
Affiliation:
Institut de Mathématiques de Jussieu, CNRS, UMR 7586, 175, rue du Chevaleret, 75013 Paris, France [email protected]
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Abstract

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This paper addresses several questions related to the Hodge conjecture. First of all we consider the question, asked by Maillot and Soulé, whether the Hodge conjecture can be reduced to the case of varieties defined over number fields. We show that this is the case for the Hodge classes whose corresponding Hodge locus is defined over a number field. We also give simple criteria for this last condition to be satisfied. Finally we discuss the relation between this condition and the notion of absolute Hodge class.

Keywords

Hodge classesHodge lociglobal invariant cycle theoremalgebraic de Rham cohomology14C3014D0714F40
Type
Research Article
Information
Compositio Mathematica , Volume 143 , Issue 4 , July 2007 , pp. 945 - 958
Copyright
Foundation Compositio Mathematica 2007