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Algebraic Cycles on Algebraic Varieties

代数簇上的代数循环

基本信息

批准号：
11440004
负责人：
SAITO Shuji
金额：
$ 5.82万
依托单位：
Nagoya University (2000-2001)Tokyo Institute of Technology (1999)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440004/
关键词：
algebraic cycles higer class field theory p-adic Hodge theory Hodge theory higher Abel-Jaoci map mixed motives Beilinson conjectures higher Chow groups logarithmic Torelli problem Hodye理論 P-進Hodye理論 Abelの定理の高次元化高次Abel-Jacobi写像 H

项目摘要

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

该研究项目有两个流程。一个是高级域理论的研究，另一个是代数圈的研究，高级域理论的目的是推广Artin-Takagi建立的经典类域理论及其应用。我们的目标是利用代数K-理论来控制算术性质的概型的交换覆盖，它可以被称为几何类场论。本文与K.加藤在此之后，该理论一直在发展，将这种新的技术，如p-adic霍奇理论本身。该研究项目的主要成果之一是推广了数论中著名的Albert-Brauer-Hasse-Noether定理，研究代数圈的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19世纪的一个里程碑式的结果。 ...更多信息 h世纪数学。的.目的是建立Abel定理的一个高维版本，即用Hodge理论分析代数簇的Chow群的结构. Griffiths在这个问题上迈出了第一步，他定义了Abel-Jacobi映射，将Chow群与复环面联系起来，称为中间Jacobian簇。然后Mumford证明了Chow群一般太大而不能被复环面控制，因此Abel-Jacobi映射可以有一个非常大的核。由此可见，Abel定理的推广问题是一个很深的问题。本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映射无法捕捉的代数圈。该理论一直在发展，并带来了各种应用Bloch的Chow群，Beilinson代数，对数Torelli问题等。Less