Algebraic Cycles on Algebraic Varieties

这个研究项目分为两部分。一个是高等场论的研究，另一个是代数循环的研究。高等类场论的目的是对Artin-Takagi所建立的经典类场论及其应用进行概括。目的是利用代数k理论控制算术性质方案的阿贝尔覆盖，可称之为几何类场论。在与K. Kato的联合工作中，建立了有理数环上有限型格式的高级场论。此后，该理论通过将p-adic Hodge理论等新技术纳入自身而不断发展。本研究项目的主要成果之一是推广了阿尔伯特-布劳尔-哈斯-诺特在数论中的著名定理。研究代数循环的一个主要目的是利用周期积分来控制代数循环。这个问题源于阿贝尔定理，这是19世纪数学界的一个里程碑式的成果。这个。目的是建立阿贝尔定理的高维版本，即利用霍奇理论分析代数变体的Chow群的结构。Griffiths迈出了解决这个问题的第一步，他定义了将Chow群与称为中间雅可比变量的复杂环面联系起来的Abel-Jacobi映射。然后Mumford证明了Chow群通常太大而不能被一个复杂的环面控制，因此Abel-Jacobi映射可以有一个非常大的核。由此认识到阿贝尔定理的推广问题是一个非常深刻的问题。本研究项目对该问题的主要贡献是构建了更高的Abel-Jacobi映射理论，推广了Griffiths的Abel-Jacobi映射，以捕获Griffiths的Abel-Jacobi映射无法捕获的代数循环。该理论一直在发展，并在Bloch’s Chow群、Beilinson猜想、对数Torelli问题等方面带来了各种应用。少

There are two streams in this research project. One is that of higher class field theory and another is that of study of algebraic cycles.The purpose of higher class field theory is to generalize the classical class field theory established by Artin-Takagi and its applications. A goal is to control abelian covering of a scheme of arithmetic nature by using algebraic K-theory and it may be called geometric class field theory. Higher class field theory for a scheme of finite type over the ring of rational integers has been established in the joint work with K. Kato. After that the theory has been developing by incorporating such new techniques as p-adic Hodge theory into itself. One of the main results of this research project generalizes the well-known theorem in number theory due to Albert-Brauer-Hasse-Noether.A main purpose of study of algebraic cycles is to control algebraic cycles by means of period integral. The problem originates from Abel's theorem, a monumental result in the 19t … More h century mathematics. The. aim is to establish a higher dimensional version of Abel's theorem, that is to analyze the structure of Chow groups of algebraic varieties by means of Hodge theory. The first step toward this problem has been taken by Griffiths, who defined Abel-Jacobi maps relating Chow group to complex torus called intermediate Jacobian variety. Then Mumford shown that Chow group is in general too large to be controlled by a complex torus and hence Abel-Jacobi map can have a very large kernel. By this result it is recognized that the problem of generalization of Abel's theorem is very deep. The main contributions of this research project to the problem is to construct the theory of higher Abel-Jacobi maps generalizing Griffiths' Abel-Jacobi maps to capture algebraic cycles that Griffiths' Abel-Jacobi maps could not captured. The theory has been developing and brought about various applications to Bloch's Chow groups, Beilinson conjectures, logarithmic Torelli problems and so on. Less