Hodge theoretic and arithmetic aspects of algebraic cycles

本文的研究包括三个部分：（I）motivic上同调的直接性（II）利用p$-adic Hodge理论研究代数圈（III）利用Hodge理论研究代数圈（I）本文给出了与（I）有关的一个结果，并报告了关于有限域上簇的Kato猜想的结果，到去年为止，我们已经通过假设奇点的分解证明了这个猜想。今年我们成功地去掉了这一假设，代之以Gabber的精化变换。算术概型的动机上同调是算术几何的一个重要研究对象。它包括代数数域的理想类群、单位群和代数簇的Chow群，它与有限域或代数数域上代数簇的L-函数有着密切的联系。其中一个重要的公开问题是猜想算术概型的动机上同调应是可生成的，它推广了算术概型的动机上同调理论。 ...更多信息 给出了代数数域的理想类群和单位群的有限性的一些已知结果。除了一维情形（即代数数域上的整数环或有限域上的曲线的情形）外，关于这个猜想的结果很少。Jannsen，我们有关的问题，以猜想加藤的非循环的某些复杂的布洛赫-Ogus型，这是一个自然的推广高维计划的哈斯原则的布劳尔群的一个全球领域，一个基本定理在数论。我们能够证明加藤猜想的品种在有限域假设决议的奇异性，这确保了一定motivic上同调有限系数的品种在有限域是finite.Recently加伯尔细化德容的结果证明存在的改变，其程度是素一个给定的素数不同的特征p的有限域。我已经成功地消除了在以前的工作与扬森的奇异性的决议的假设，以显示素的一部分，p的加藤猜想无条件。少

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