On the locus of Hodge classes

Abstract:

Let $S$ be a nonsingular complex algebraic variety and $\mathcal {V}$ a polarized variation of Hodge structure of weight $2p$ with polarization form $Q$. Given an integer $K$, let ${S^{(K)}}$ be the space of pairs $(s,u)$ with $s \in S$, $u \in {\mathcal {V}_s}$ integral of type $(p,p)$, and $Q(u,u) \leq K$. We show in Theorem 1.1 that ${S^{(K)}}$ is an algebraic variety, finite over $S$. When $\mathcal {V}$ is the local system ${H^{2p}}({X_s},\mathbb {Z})$/torsion associated with a family of nonsingular projective varieties parametrized by $S$, the result implies that the locus where a given integral class of type $(p,p)$ remains of type $(p,p)$ is algebraic.