Hodge theoretic and arithmetic aspects of algebraic cycles

代数循环的霍奇理论和算术方面

• 批准号: 18340003
• 负责人: SAITO Shuji
• 金额: $ 6.53万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2006
• 资助国家: 日本
• 起止时间: 2006 至 2009
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18340003/
• 关键词: 代数的サイクル; モチフィックコホモロジー; Chow群; 高次Chow軍; Abelの定理; Hodge理論; p進Hodge理論; 高次元類体論; ススリンホモロジー; エタールコホモロジー; ホッヂ理論; p-進ホッヂ理論; 有限性; 数論的多様体; レギュレーター写像; P進Hodge理論; 混合Hodge加群; モチフィツクコホモロジー; Hodge構造の無限小変動; 加藤予想; 有限性定理; Hasse原理; Bloch-Ogusの理論; ホモロジ

The research consists of three parts :(I) Finiteness of motivic cohomology(II) Study of algebraic cycles by using the $p$-adic Hodge theory(III) Study of algebraic cycles by using the Hodge theory.In what follows we explain a result related to (I).We report on our result on the Kato conjecture for varieties over finite fields.By the last year, we have proved the conjecture by assuming resolution of singularities. This year we succeeded in removing the assumption by replacing it with Gabber's refined alteration.Motivic cohomology of arithmetic schemes is an important object to study in arithmetic geometry. It includes the ideal class group and the unit group of an algebraic number field, and the Chow groups of algebraic varieties.It is closely related to the L-functions of algebraic varieties over a finite field or an algebraic number field. One of the important open problem is the conjecture that motivic cohomology of arithmetic schemes should be finitely generated, which generalizes t … More he known finiteness results on the ideal class group and the unit group of an algebraic number field. There has been only few results on the conjecture except the one-dimensional case (namely the case of integer rings of an algebraic number field or curves over a finite field).In a joint work with U. Jannsen we related the problem to a conjecture of Kato on the acyclicity of a certain complexes of Bloch-Ogus type, which is a natural generalization to higher dimensional schemes of the Hasse principle for the Brauer group of a global field, a fundamental theorem in number theory. We were able to prove the Kato conjecture for varieties over finite fields assuming resolution of singularities, which ensured that certain motivic cohomology with finite coefficient of varieties over finite fields is finite.Recently Gabber refined de Jong's result by proving the existence of an alteration whose degree is prime to a given prime different from the characteristic p of the finite field. I have succeeded in removing the assumption of resolution of singularities in the previous work with Jannsen to show the prime-to-p part of the Kato conjecture unconditionally. Less

本研究包括三个部分：(1)动机上同调的有限性；(2)用$p$-adic Hodge理论研究代数环；(3)用Hodge理论研究代数环。下面我们将解释与(1)有关的一个结果。我们报告了有限域上变分的加藤猜想的结果。到去年为止，我们已经通过假设奇点的分辨率证明了这个猜想。今年我们成功地去掉了这个假设，代之以Gabber的精细化修改。算术格式的动机上同调是算术几何中一个重要的研究对象。它包括代数数域的理想类群和单位群，以及代数变种的周氏群。它与有限域或代数数域上的代数变异的l函数密切相关。其中一个重要的开放问题是算术格式的动机上同调是有限生成的猜想，它推广了代数数域的理想类群和单位群的已知有限性结果。除了一维情况（即代数数域上的整数环或有限域上的曲线）外，关于这个猜想的结果很少。在与U. Jannsen合著的一篇文章中，我们将该问题与Kato关于Bloch-Ogus型复合体的非环性的猜想联系起来，这是对全局域Brauer群的Hasse原理的高维格式的自然推广，是数论中的一个基本定理。在奇点分解的前提下，证明了有限域上变项的Kato猜想，从而保证了有限域上变项的有限系数的动机上同调是有限的。最近，Gabber改进了de Jong的结果，证明了一个变化的存在性，它的度是素数到一个给定素数，不同于有限域的特征p。我已经成功地取消了先前与Jannsen合作的工作中关于奇点分解的假设，从而无条件地证明了加藤猜想的素数到p部分。少

项目成果

A $p$-adic regulator map and finiteness results for arithmetic schemes

算术方案的 $p$-adic 调节器映射和有限性结果

• 发表时间: 2010
• 期刊: Documenta Math.Extra Volume : Andrei A.Suslin's Sixtieth Birthday
• 作者: S.Saito;K.Sato
• 通讯作者: K.Sato

Beilinson's Hodge conjecture with coefficient for open complete intersections

具有开完全交集系数的 Beilinson 霍奇猜想

• 发表时间: 2007
• 期刊: the proceedings of Workshop Algebraic Cycles and Motives, EAGER Conerence 2004 in Leiden in honor of Prof. J. Murre.. (To appear)
• 作者: M.Asakura;S.Saito
• 通讯作者: S.Saito

Finiteness of motivic cohomology, cohomological Hasse principle special values of zeta functions

动机上同调的有限性、上同调哈斯原理 zeta 函数的特殊值

• 发表时间: 2009
• 作者: M.Asakura;S.Saito;斎藤毅;Takeshi Saito;斎藤毅;Takeshi Saito;Tomohide Terasoma;Shuji Saito
• 通讯作者: Shuji Saito

代数学II環上の加群

代数 II 环上的模块

• 发表时间: 2007
• 作者: 倉持清美;金子京子;阿部睦子;妹尾理子;望月一枝;T. Satoh;Shuji Saito;倉持清美・伊藤葉子・岡野雅子・金田利子;Takakazu Satoh;Shuji Saito;N. Kurokawa;岡野雅子・伊藤葉子・倉持清美・金田利子;桂 利行
• 通讯作者: 桂 利行

Maximal components of Noether-Lefschetz locus for Beilinson-Hodge cycles

Beilinson-Hodge 循环的 Noether-Lefschetz 轨迹的最大分量

• 发表时间: 2008
• 期刊: Math. Ann. 341
• 作者: Saito;S.;Asakura;M.
• 通讯作者: M.