Algebraic cycles, normal functions and iterated integrals

代数循环、正规函数和迭代积分

• 批准号: 0703956
• 负责人: Gregory Pearlstein
• 金额: $ 10.57万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0703956&HistoricalAwards=false
• 关键词: Algebraic; cycles; normal; functions; iterated

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.

该提案有两个主要目标。首先是证明复代数簇上的容许正规函数的零轨迹是代数的。这种正规函数与 Hodge 猜想相关，如下所示：设 X 是维度为 2n 的平滑复射影簇，L 是 X 上的一个非常充足的线丛。然后，X 上的原始 Hodge 类定义了一个在附属于 L 的完整线性系统的 Zariski 开子集。第二个目标是建立关于切向基点的光滑复代数簇的基本群的相对完备的极限周期的存在性，并使用结果研究模形式的迭代积分。这两个项目的统一主题是研究混合 Hodge 结构变化的渐近行为。周期积分是代数函数在代数集上积分的推广。长期以来，此类积分在几何、数论和物理学中一直很重要。通过允许被积函数和/或积分域以适当的方式取决于参数，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函数。尽管如此，代数几何（正规函数）中出现的某些周期积分系统的零轨迹预计是代数的。该提案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 博士计划利用周期积分的渐近行为来研究数论中出现的某些重要函数的特殊值。