Algebraic Cycles and Normal Functions

About This Grant

The purpose of this project is to bring techniques from Hodge theory to bear on the properties of solutions of differential equations found throughout mathematics and physics. To elaborate, algebraic geometry is the study of systems of polynomial equations. Hodge theory seeks to understand the shape of the algebraic spaces defined by their solution sets. In so doing, it produces powerful connections between algebraic geometry and other parts of mathematics and physics, such as number theory, differential equations, and string theory. Normal functions are solutions to differential equations arising from algebraic cycles (or subspaces) on an algebraic space. By recognizing a solution as a normal function, which is a nontrivial task, one gets access to its behavior at numerous "special" points. In the scenarios to be considered in this project, these "special values" have important applications to identities and conjectures in number theory, the classification of algebraic spaces such as Fano varieties and algebraic curves, and the spectra of operators in topological string theory. This project will also help to train the next generation of researchers in pure mathematics, by integrating the PI's graduate students in work on specific problems in Hodge theory and bringing outside consultants to Washington University. Results will be disseminated through conferences, summer schools, journal articles, and websites. Normal functions originated in the work of Poincare and Lefschetz in the early 20th century. They are to families of algebraic cycles what period maps are to families of algebraic varieties: the basic Hodge-theoretic invariant. Informally, they are given by integrals of differential forms on non-closed chains instead of topological cycles, and satisfy inhomogeneous differential equations instead of homogeneous ones. The PI will study the properties, invariants, classification, and geometric realization of normal functions. The settings and applications range from arithmetic geometry and number theory to mirror symmetry, moduli, and mathematical physics. The specific goals of this project are to: (I) produce new evidence for the Beilinson and Bloch-Kato conjectures on special values of L-functions, and provide motivic realizations of biextensions; (II) describe an A-model variation of mixed Hodge structure on the cohomology of a Fano variety, and a B-model Lefschetz principle governing normal functions on LG-models; and (III) classify and realize normal functions with values in Hermitian and hypergeometric variations of Hodge structure, with applications to moduli of curves, hypergeometric identities, and quantization. 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.