Cattani, EDELIGNE, PKAPLAN, A2024-04-262024-04-261995https://hdl.handle.net/20.500.14394/34511<p>This is the pre-published version harvested from ArXiv. The published version is located at <a href="http://www.jstor.org/stable/2152824">http://www.jstor.org/stable/2152824</a></p>Let S be a nonsingular complex algebraic variety and 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 ∈ S, u ∈ Vs integral of type (p, p), and Q(u, u) ≤ K. We show in Theorem 1.1 that S(K) is an algebraic variety, finite over S. When V is the local system H2p (Xs, 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.article