2009 Fiscal Year Final Research Report
Hodge theoretic and arithmetic aspects of algebraic cycles
| Project/Area Number |
18340003
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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 | The University of Tokyo |
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Principal Investigator |
SAITO Shuji The University of Tokyo, 大学院・数理科学研究科, 教授 (50153804)
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| Project Period (FY) |
2006 – 2009
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| Keywords | 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 |
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| Research Abstract |
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
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