Singularities of Normal Functions and Algebraic Cycles

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This proposal has three main objectives. The first is to extend the results of Cattani, Deligne and Kaplan on the algebraicity of the locus of a Hodge class in a variation of pure Hodge structure to variation of mixed Hodge structures. The second is to study the asymptotics behavior of the archimedean height of a family of cycles and the singularities of the associated normal functions in the sense of Phillip Griffiths and Mark Green. The third objective is the study of the limiting periods of the relative completion of the fundamental group of a smooth complex algebraic variety with respect to tangential base points. A period integral is a generalization of the integral of an algebraic function over an algebraic set. Such period integrals have long been of importance in number theory, geometry and physics. By allowing the integrand and/or domain of integration to vary, such period integrals define holomorphic functions of the parameters. The object of this proposal is to shed light on the Hodge conjecture and other basic questions in algebraic geometry via the study of the asymptotic behavior of such functions. In addition to pure mathematics, such period integrals play an impor­tant part in a branch of physics called string theory, which seeks to unify gravity and quantum mechanics.