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

University Of Chicago, Chicago IL

Investigators

Abstract

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.