Hodge loci and absolute Hodge classes

Details Hodge loci and absolute Hodge classes Hodge loci and absolute Hodge classes

2006-05-30

arxiv.org/abs/math/0605766v1

Mathematics

Algebraic Geometry

Scientific paper

We combine Deligne's global invariant cycle theorem, and the algebraicity theorem of Cattani, Deligne and Kaplan, for the connected components of the locus of Hodge classes, to conclude that under simple assumptions these components are defined over number fields (assuming the initial family is), as expected from the Hodge conjecture. We also show that the Hodge conjecture for (weakly) absolute Hodge classes reduces to the Hodge conjecture for (weakly) absolute Hodge classes on varieties defined over number fields.

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