Classifying spaces of degenerating Hodge structures, the p-adic analogue, and related arithmetic study

退化 Hodge 结构的分类空间、p-adic 类似物以及相关算术研究

• 批准号: 1001729
• 负责人: Kazuya Kato
• 金额: $ 42.71万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001729&HistoricalAwards=false
• 关键词: Classifying; spaces; degenerating; Hodge; structures

In this proposal, the principal investigator K. Kato intends to study degenerations of Hodge structures and related problems, collaborating with the co-principal investigators S. Bloch and T. Fukaya. K. Kato and S. Usui constructed toroidal partial compactifications of classifying spaces of polarized Hodge structures in which points at infinity correspond to degenerations of Hodge structures. He is now generalizing this theory to treat mixed Hodge structures and also p-adic Hodge structures. With S. Bloch, he plans to study asymptotic behaviors of period integrals and regulators in degeneration, by using the study of degenerations of Hodge structures. Period integrals and regulators are related to values of zeta functions. The PI's intend to study related problems concerning arithmetic properties of zeta values. It is expected that the study of this proposal on degenerations of Hodge structures have various applications. For example, Hodge conjecture is related to degeneration of intermediate Jacobian which is understood as a class group of degenerating Hodge structures, and so, it is expected that the study of this proposal can contribute to the solution of Hodge conjecture. S. Bloch studied the relation of the theory of motives and the period integrals which appear in physics. The divergence of such period integral is an important subject in physics, and it is expected that the divergence is well understood by the degeneration of motives and degeneration of associated Hodge structures. The study of this proposal is expected to have applications to physics.Understanding of degeneration for geometric objects (spaces or mathematical structures) is an important but difficult problem. By constructing enlarged classifying spaces of mathematical structures in which points on the boundary correspond to degenerations, the PI's can better understand degeneration. For example, in physics it is important to understand various infinite limits. In this program, the divergences in physics are understood as arising from degeneration of mathematical structures, and it is expected that this study will clarify such asymptotic behavior. Values of zeta functions often appear in physics, and K. Kato, S. Bloch and T. Fukaya have studied arithmetic properties of zeta values. They intend to study the relations between degenerations and zeta values. One may hope that in this way the deep relation between nature and arithmetic can be better understood. More generally, many unsolved problems in mathematics are related to degeneration, and it is expected that the study of this proposal will contribute to the solution of them.

在该提案中，首席研究员 K. Kato 打算与联合首席研究员 S. Bloch 和 T. Fukaya 合作，研究 Hodge 结构的退化及相关问题。 K. Kato 和 S. Usui 构造了极化 Hodge 结构分类空间的环形部分紧化，其中无穷远点对应于 Hodge 结构的退化。他现在正在推广这一理论来处理混合 Hodge 结构以及 p-adic Hodge 结构。他计划与 S. Bloch 一起，通过霍奇结构的简并研究，研究简并过程中周期积分和调节子的渐近行为。周期积分和调节器与 zeta 函数的值相关。 PI打算研究有关zeta值算术性质的相关问题。 预计这项关于 Hodge 结构退化的提议的研究具有多种应用。例如，霍奇猜想与中间雅可比行列式的退化有关，中间雅可比行列式被理解为退化霍奇结构的类群，因此，期望该提议的研究能够有助于霍奇猜想的解决。 S.布洛赫研究了动机理论与物理学中出现的周期积分之间的关​​系。这种周期积分的散度是物理学中的一个重要课题，并且期望通过动机的简并和相关霍奇结构的简并来很好地理解该散度。该提议的研究有望应用于物理学。理解几何对象（空间或数学结构）的退化是一个重要但困难的问题。通过构建数学结构的扩大分类空间，其中边界上的点对应于退化，PI 可以更好地理解退化。例如，在物理学中，理解各种无限极限非常重要。在这个程序中，物理学中的分歧被理解为由数学结构的退化引起的，并且预计这项研究将澄清这种渐近行为。 zeta函数的值经常出现在物理学中，K. Kato、S. Bloch和T. Fukaya研究了zeta值的算术性质。他们打算研究退化和 zeta 值之间的关系。人们可能希望通过这种方式可以更好地理解自然与算术之间的深层关系。更一般地说，数学中许多未解决的问题都与退化有关，预计该提案的研究将有助于解决这些问题。