Fields of Definition of Hodge Loci

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...

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 ...

Hodge loci and absolute Hodge classes

Eigenvalues of Frobenius and Hodge Numbers

We study the connection between Hodge purity of the cohomology of algebraic varieties over fields of different characteristics. Specifically, we study varieties over number fields, whose cohomology in some fixed degree 2i consists entirely of Hodge classes, that is, whose Hodge cohomology in degree 2i is entirely of type (i, i). Among other things, we give sufficient conditions for varieties ov...

One-form Abelian Gauge Theory as the Hodge Theory

We demonstrate that the two (1 + 1)-dimensional (2D) free 1-form Abelian gauge theory provides an interesting field theoretical model for the Hodge theory. The physical symmetries of the theory correspond to all the basic mathematical ingredients that are required in the definition of the de Rham cohomological operators of differential geometry. The conserved charges, corresponding to the above...