Hodge theoretic and arithmetic aspects of algebraic cycles

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

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