Algebraic Cycles on Algebraic Varieties

这项研究项目有两个流派。高类场理论的目的是推广Artin-Takagi建立的经典类场论及其应用。一个目标是利用代数K-理论来控制算术性质的方案的阿贝尔覆盖，这可以称为几何类域理论。在与K.Kato的合作中，建立了有理整数环上有限类型格式的高类场理论。在此之后，该理论通过融入p-进Hodge理论等新技术而得到发展。这一研究项目的主要成果之一是推广了阿尔伯特-布劳尔-哈斯-诺瑟的数论中著名的定理。研究代数圈的一个主要目的是通过周期积分来控制代数圈。这个问题起源于阿贝尔定理，这是19T…中的一个不朽的结果更多的h世纪数学。这个。目的是建立Abel定理的高维版本，即利用Hodge理论分析代数簇的Chow群的结构。这个问题的第一步是由Griffiths迈出的，他定义了将Chow群与称为中间Jacobian簇的复杂环面联系起来的Abel-Jacobi映射。然后Mumford证明了Chow群通常太大而不能由一个复杂的环面控制，因此Abel-Jacobi图可以有一个非常大的核。从这一结果可以看出，Abel定理的推广问题是非常深刻的。这一研究项目的主要贡献是构造了高阶Abel-Jacobi映射理论，推广了Griffiths的Abel-Jacobi映射，以捕捉Griffiths的Abel-Jacobi映射不能捕获的代数圈。该理论一直在发展，并在Bloch的Chow群、Beilinson猜想、对数Torelli问题等方面得到了广泛的应用。较少

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