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

在这项提案中，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结构的退化和相关问题。K. Kato和S. Usui构造了极化Hodge结构分类空间的环面部分紧化，其中无穷远处的点对应于Hodge结构的退化。他现在将这一理论推广到混合Hodge结构和p进Hodge结构。与S. Bloch一起，他计划利用Hodge结构的退化研究周期积分和退化中的调节器的渐近行为。周期积分和调节器与ζ函数的值有关。PI打算研究有关zeta值的算术性质的相关问题。这一建议对霍奇结构退化的研究具有广泛的应用前景。例如，Hodge猜想与中间雅可比矩阵的退化有关，中间雅可比矩阵被理解为退化的Hodge结构的一类群，因此，期望该建议的研究能够有助于Hodge猜想的求解。布洛赫研究了物理学中出现的动机理论和周期积分的关系。这种周期积分的散度是物理学中的一个重要问题，期望通过动机的退化和相关霍奇结构的退化来很好地理解散度。对这一建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）的退化是一个重要而又困难的问题。通过构造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，理解各种无限极限是很重要的。在这个程序中，物理上的分歧被理解为由数学结构的退化引起的，期望本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刻关系。更一般地说，数学中许多未解决的问题都与退化有关，期望本提案的研究将有助于解决这些问题。

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.