This proposal has two main objectives. The first is to prove that the zero locus of an admissible normal function on a complex algebraic variety is algebraic. Such normal functions arise in connection with the Hodge conjecture as follows: Let X be a smooth complex projective variety of dimension 2n and L be a very ample line bundle over X. Then, a primitive Hodge class on X defines an admissible normal function over a Zariski open subset of the complete linear system attached to L. The second objective is to establish the existence of the limiting periods of the relative completion of the fundamental group of a smooth complex algebraic variety with respect to tangential base points and use the results to study iterated integrals of modular forms. The unifying theme between these two projects is the study of the asymptotic behavior of variations of mixed Hodge structure.
A period integral is a generalization of the integral of an algebraic function over an algebraic set. Such integrals have long been of importance in geometry, number theory and physics. By allowing the integrand and/or domain of integration to depend upon parameters in an appropriate way, such period integrals define holomorphic functions on the parameter space. In general, these period functions are not algebraic. Nonetheless, the zero locus of certain systems of period integrals arising in algebraic geometry (normal functions) are expected to be algebraic. The first goal of this proposal is to study the asymptotic behavior of period functions, and use the results to prove the algebraicity of the zero locus of a normal function. In a similar spirit, Dr. Pearlstein plans to use the study the asymptotic behavior of period integrals to study the special values of certain important functions arising in number theory.
