Complex Geometric and Lie Theoretic Aspects of Hodge Theory

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Algebraic geometry is a field of mathematics concerned with the solutions of (systems of) polynomial equations. These solution sets are called algebraic varieties, and one is interested in understanding their algebraic and geometric properties. These structures can be quite complicated and mysterious. One way to gain some insight is to consider the variety's "Hodge structure". This is a simpler object with a linear algebraic nature, that nonetheless often encodes a great deal of information about the variety. The theme of this project is the application of representation theory (a sophisticated generalization of linear algebra) to obtain geometric insight into these varieties.Topics addressed in this project include: (a) degenerations of Hodge structure, and their application to the study of moduli spaces and their compactifications; (b) characteristic varieties of a variation of Hodge structure (this program is driven by a close analogy with the Hwang-Mok program to study Fano manifolds via their varieties of minimal rational tangents); and (c) structure of hyperkahler cohomology as a representation of the Looijenga-Lunts-Verbitsky algebra.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.