Algebraic cycles, normal functions and iterated integrals

这项提议有两个主要目标。首先证明了复代数变量上的可容许正规函数的零轨迹是代数的。与霍奇猜想有关的正规函数如下：设X是维数为2n的光滑复射影变化，L是X上的一个非常充裕的线束，X上的一个原始Hodge类定义了l上的完全线性系统的Zariski开子集上的一个可容许正规函数。第二个目的是建立光滑复代数变量的基本群相对于切基点的完备性的极限周期的存在性，并利用这一结果研究模形式的迭代积分。这两个项目之间的统一主题是混合Hodge结构变化的渐近行为的研究。周期积分是代数函数在代数集合上的积分的推广。这种积分在几何学、数论和物理学中一直很重要。通过允许被积和/或积分的定义域以适当的方式依赖于参数，这样的周期积分定义了参数空间上的全纯函数。一般来说，这些周期函数不是代数的。尽管如此，在代数几何（正规函数）中出现的某些周期积分系统的零轨迹被期望是代数的。本文的第一个目标是研究周期函数的渐近行为，并利用结果证明正规函数零轨迹的代数性。本着类似的精神，Pearlstein博士计划利用对周期积分的渐近行为的研究来研究数论中出现的某些重要函数的特殊值。

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.