On the Locus of Hodge Classes

Abstract

Let $f:X\to S$ be a family of non singular projective varieties parametrized by a complex algebraic variety $S$. Fix $s\in S$, an integer $p$, and a class $h\in {\mathrm{H}}^{2p}(X_s,\mathbb{Z})$ of Hodge type $(p,p)$. We show that the locus, on $S$, where $h$ remains of type $(p,p)$ is algebraic. This result, which in the geometric case would follow from the rational Hodge conjecture, is obtained in the setting of variations of Hodge structures.

25 pages, Plain TeX