On the Locus of Hodge Classes

Cattani, Eduardo; Deligne, Pierre; Kaplan, Aroldo

On the Locus of Hodge Classes

Authors

Cattani, Eduardo

Deligne, Pierre

Kaplan, Aroldo

Description

Let $f: X \rightarrow S$ 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 $h \in {\rm H}^{2p}(X_s,\Z)$ 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