Singularities of Normal Functions and Algebraic Cycles

这项提议有三个主要目标。一是将Cattani，Deligne和Kaplan关于纯Hodge结构变形中Hodge类轨迹的代数性的结果推广到混合Hodge结构变形中。第二个是在Phillip Griffiths和Mark Green意义下研究一族圈族的阿基米德高度的渐近行为和伴随的正规函数的奇异性。第三个目标是研究光滑复代数簇的基本群相对于切向基点的相对完备性的极限周期。周期积分是代数函数在代数集合上积分的推广。这种周期积分在数论、几何学和物理学中一直很重要。通过允许被积函数和/或积分区域的变化，这样的周期积分定义了参数的全纯函数。这个建议的目的是通过对这些函数的渐近行为的研究来阐明Hodge猜想和代数几何中的其他基本问题。除了纯数学之外，这样的周期积分在物理学的一个叫弦理论的分支中也扮演着重要的角色，该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.