The Geometry of Quasi-modular Forms

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This award supports research in the field of algebraic geometry, which deals with nonlinear shapes defined by polynomial equations in many variables. To better understand their geometry, one considers potential functions and force fields (differential forms) on these shapes, taking a cue from Einstein’s approach to physics on a curved spacetime. Force fields are linear objects, and thus can be studied using methods of linear algebra and group theory. Among the more sophisticated methods from this point of view are modular forms. This project will explore the constraints placed on smooth nonlinear shapes from modular forms, and then which features persist when the shapes acquire singularities. To describe singular shapes, we must introduce more general tools called quasi-modular and mock modular forms. The project has several avenues for participation by undergraduate and graduate student researchers, as well as fruitful connections with the neighboring fields of symplectic geometry and number theory. The project addresses central questions in several areas of complex algebraic geometry, beginning with Hodge theory and extending through enumerative geometry, mirror symmetry, and Severi varieties. These areas are tied together by the appearance of modular forms and their generalizations: Siegel, quasi, and mock. A broad theme is the importance of K-trivial varieties, most notably elliptic curves. The investigator will vastly generalize a theorem of Borcherds about cycle-valued modular forms on locally symmetric spaces, and then explore an array of geometric implications. There is an intriguing analogy between completed Shimura varieties and Kontsevich spaces of stable maps. A new Torelli theorem for elliptic surfaces leads to a system of conjectural correspondences which interchange degree and genus.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.