Singularities of Normal Functions and Algebraic Cycles

这项建议有三个主要目标。第一个是将Cattani，Deligne和Kaplan关于纯Hodge结构的变分中Hodge类的轨迹的代数性的结果推广到混合Hodge结构的变分中。第二是研究了圈族的阿基米德高度的渐近性态以及相应正规函数在菲利普格里菲斯和马克绿色意义下的奇点。第三个目的是研究光滑复代数簇的基本群关于切基点的相对完备化的极限周期。周期积分是代数函数在代数集合上的积分的推广。这样的周期积分在数论、几何和物理学中一直很重要。通过允许被积函数和/或积分域变化，这样的周期积分定义了参数的全纯函数。这一建议的目的是阐明霍奇猜想和其他基本问题的代数几何通过研究的渐近行为，这样的功能。除了纯数学之外，这种周期积分在物理学的一个分支--弦理论中也扮演着重要的角色，弦理论试图统一引力和量子力学。

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.