The PI proposes to apply methods from the theory of mixed Hodge modules to the study of Hodge classes and normal functions, in order to solve several problems about singularities of normal functions and the locus of Hodge classes. The main idea is the construction of natural complex-analytic extension spaces, which make it possible to apply differential-geometric and topological methods to the above problems, and which can also be used to define numerical invariants. To date, the PI has successfully carried out one portion of this, namely the construction of complex-analytic Neron models. He proposes to use his results to describe graph closures of admissible normal functions, to carry out a local study of singularities, and to obtain a classification theorem. The PI also proposes to introduce Poincare bundles of Neron models, both to define numerical invariants for pairs of normal functions and to make progress on the question of the existence of singularities.
The proposed work is connected with the program of M. Green and P. Griffiths, and bears on the Hodge conjecture. The general philosophy is to understand the geometry of a complex algebraic variety with the help of differential geometry and topology. The proposed methods can be used, for instance, to study Noether-Lefschetz loci on Calabi-Yau threefolds; this example is interesting because we have very intriguing numerical predictions made by physicists for these loci.
