The extended locus of Hodge classes

We introduce an "extended locus of Hodge classes" that also takes into account integral classes that become Hodge classes "in the limit". More precisely, given a polarized variation of integral Hodge structure of weight zero on a Zariski-open subset of a complex manifold, we construct a canonical analytic space that parametrizes limits of integral classes; the extended locus of Hodge classes is an analytic subspace that contains the usual locus of Hodge classes, but is finite and proper over the base manifold. The construction uses Saito's theory of mixed Hodge modules and a small generalization of the main technical result of Cattani, Deligne, and Kaplan. We study the properties of the resulting analytic space in the case of the family of hyperplane sections of an odd-dimensional smooth projective variety.