Hodge loci and absolute Hodge classes

Hodge loci and absolute Hodge classes

0 M ay 2 00 6 Hodge loci and absolute Hodge classes Claire Voisin

Let π : X → T be a family of smooth projective complex varieties. Assume X , π, T are defined over Q. An immediate consequence of the fact that there are only countably many components of the relative Hilbert scheme for π, and that the relative Hilbert scheme (with fixed Hilbert polynomial) is defined over Q, is the following: if the Hodge conjecture is true, the components of the Hodge locus i...

Notes on absolute Hodge classes

Absolute Hodge classes first appear in Deligne’s proof of the Weil conjectures for K3 surfaces in [14] and are explicitly introduced in [16]. The notion of absolute Hodge classes in the singular cohomology of a smooth projective variety stands between that of Hodge classes and classes of algebraic cycles. While it is not known whether absolute Hodge classes are algebraic, their definition is bo...

Hodge loci

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 ...

The Extended Locus of Hodge Classes

We introduce an “extended locus of Hodge classes” that also takes into account integral classes that become Hodge classes “in the limit”. More precisely, given a polarized variation of integral Hodge structure of weight zero on a Zariski-open subset of a complex manifold, we construct a canonical analytic space that parametrizes limits of integral classes; the extended locus of Hodge classes is...