In a family of projective complex algebraic varieties, all nonsingular fibers are topologically equivalent; in particular, their cohomology groups are isomorphic. Near the “boundary,” where the varieties acquire singular points, this is no longer the case.
The theory of variations of Hodge structure provides strong tools to understand the local behavior near points on the boundary; these have been used, for instance, to prove that the locus of Hodge classes is a union of algebraic varieties (by Cattani, Deligne, and Kaplan).
Recently, there has been interest in global questions related to the behavior at the boundary, especially for the family of all hypersurfaces (of large degree) of a given smooth projective variety. Green and Griffiths introduced the concept of the “singularity” of a normal function; following their ideas,
Brosnan, Fang, Nie, and Pearlstein, and de Cataldo and Migliorini proved that the Hodge conjecture is equivalent to the existence of such singularities.
In this dissertation, we investigate the boundary behavior of cohomology classes in families (in the above sense), from several different points of view. We also obtain new interpretations for the singularity of a normal function in the family of hypersurface sections of sufficiently large degree.