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

该研究项目中有两个流。一种是较高阶级的田地理论，另一个是对代数周期的研究。高级领域理论的目的是概括Artin-Takagi及其应用所建立的经典阶级领域理论。一个目标是通过使用代数K理论来控制算术性质方案的Abelian覆盖，并且可以称为几何类阶级理论。在与K. Kato的联合工作中建立了有限类型方案的高级野外理论。之后，该理论通过将诸如P-Adic Hodge理论之类的新技术纳入自身来发展。该研究项目的主要结果之一是，由于Albert-Brauer-Hasse-Noether的数字理论，众所周知的定理。代数周期研究的主要目的是通过周期积分来控制代数周期。该问题起源于亚伯定理，这是19t的每月结果……更多的h世纪数学。目的是建立Abel定理的较高维度版本，即通过Hodge理论分析代数变化的Chow群体的结构。格里菲斯（Griffiths）迈出了这一问题的第一步，格里菲斯（Griffiths）将Abel-Jacobi地图定义为Chow group与复杂的圆环（称为中间Jacobian品种）。然后，芒福德（Mumford）表明，乔集团（Chow Group）通常太大而无法由复杂的圆环控制，因此亚伯 - 雅各比（Abel-Jacobi）图可能具有很大的内核。通过这种结果，人们认识到，亚伯定理的概括问题非常深。该研究项目对该问题的主要贡献是构建较高的亚伯 - 雅各比（Abel-Jacobi）地图，概括了格里菲斯（Griffiths）的亚伯 - 雅各比（Abel-Jacobi）地图，以捕获格里菲斯（Griffiths）的Abel-Jacobi地图无法捕获的代数周期。该理论一直在开发并为Bloch的Chow群体，Beilinson猜想，对数Torelli问题等提供了各种应用。较少的