2001 Fiscal Year Final Research Report Summary
Algebraic Cycles on Algebraic Varieties
| Project/Area Number |
11440004
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Nagoya University (2000-2001)
Tokyo Institute of Technology (1999) |
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Principal Investigator |
SAITO Shuji Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (50153804)
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| Co-Investigator(Kenkyū-buntansha) |
KOBAYASHI Ryoichi Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (20162034)
MATSUMOTO Koji Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (60192754)
FUJIWARA Kazuhiro Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (00229064)
KONDO Shigeyuki Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (50186847)
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| Project Period (FY) |
1999 – 2001
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| Keywords | 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 |
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| Research Abstract |
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
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