  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.
