Hodge loci
نویسنده
چکیده
The goal of this expository article is first of all to show that Hodge theory provides naturally defined subvarieties of any moduli space parameterizing smooth varieties, the “Hodge loci”, although only the Hodge conjecture would guarantee that these subvarieties are defined on a finite extension of the base field. We will show how these subsets can be studied locally in the Euclidean topology and introduce a number of related Hodgetheoretic notions. The article will culminate with two results by Deligne, Cattani-Deligne-Kaplan respectively. The first one says that Hodge classes are absolute Hodge on abelian varieties. This is a statement which we will rephrase in general in terms of Hodge loci and is enough to guarantee that Hodge loci are closed algebraic, defined on a finite extension of the base field. The second tells us that Hodge loci are in general closed algebraic, as predicted by the Hodge conjecture.
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