Complex Geometric Properties of Period Maps

Duke University, Durham NC

Investigators

Abstract

Algebraic geometry studies geometric properties of sets that arise as solutions to systems of polynomial equations. Many familiar sets arise in this way, including lines and planes, and circles and spheres/soap bubbles. Algebraic geometry also has many applications outside of mathemayics, including statistics, robotics, error-correcting codes, phylogenetics, game theory and integer programming. “Moduli problems” is a sub-branch of algebraic geometry that studies these sets in families. For example, one might consider all the lines passing through a fixed point. These families, and the related questions, can be quite complicated. An important tool in their study is the “period map”, a function that is associated with the family. The period map allows one to bring to bear techniques from other areas of mathematics, including complex geometry (where the defining functions are not polynomials, as in algebraic geometry, but holomorphic functions) and representation theory (a sophisticated extension of linear algebra). This award supports both: (i) research into properties of period maps that are motived by applications to moduli problems, and (ii) the training of graduate students and early career postdoctoral researchers. The PI’s work includes active mentorship of both undergraduate and graduate students through multiple departmental and university programs. Half of the projects aim to construct completions of period maps, motivated by anticipated applications to compactifications of moduli spaces. This includes an Ohsawa-Takegoshi type extension problem that will complete a program to generalize the Satake-Baily-Borel compactification; an investigation of this construction for a specific family of Calabi-Yau varieties; and the construction of a Kato-Usui type completions for various period maps, including that associated to a specific family of Calabi-Yau 3-folds. The other projects continue a program initiated by the PI to study period maps (and variations of Hodge structure) by their characteristic forms (which are infinitesimal, complex-geometric invariants associated with a period maps). This program is inspired by a close analogy with the very successful Hwang-Mok program to study Fano manifolds via their varieties of minimal rational tangents. 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.