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Period integral of algebraic varieties

代数簇的周期积分

基本信息

批准号：
16540011
负责人：
TERASOMA Tomohide
金额：
$ 2.46万
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 (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16540011/
关键词：
motif bar construction arithmetic geometric mean Hodge structures 周期積分多重ゼータ値

项目摘要

(1) We define a homomorphism from the polylog complex defined by Goncharov to the extension group of mixed Tate motives defined by Bloch and Kriz. The existence of this map is conjectured by Beilinson-Deligne. Beilinson and Deligne assume some conjectures which are equivalent to the KP1 conjecture and Beilinson-Soule conjecture. We constructed the homomorphism without assuming these conjecture using bar constructions and recovery principle for them.(2) We show the differential graded category equivalence between the category of comodules of differential graded Hopf algebra and that of differential graded complex over differential graded category associated a differential graded algebra Using this equivalence, we showed that the Hopf coalgebra constructed form the Deligne algebra classifies the category of variation of mixed Tate Hodge structures.(3) We give a description of the pro-p completion of algebraic varieties of positive characteristic, which classifies the Tannakian category o … More f Fp local systems in terms of Bar constructions. To get a product structure on the corresponding Hopf algebra, we introduce a homotopy shuffle product. Via this shuffle product, we get a notion of group like elements which give the description of pro-p completion.(4) We define arithmetic-geometric mean for hyperelliptic cureves of higher genus, which is a generalization of Gauss arithmetic geometric mean- Moreover, we showed that they coincides with certain determinant of periods of hyperellipitic curves using Thomae's formlula. We showed that this is also equal to a period of certain Calabi-Yau varieties.(5) We construct certain algebraic correspondence between Jacobian varieties and Calabi-Yau varieties obtained by a double covering of three dimensional projective space branched at the projective dual of Caylay Octad of genus three cureves. More over the third cohomology of the Calabi-Yau varieties are not exterior product in general by looking infinitesimal variation of Hodge structures. Less

(1)我们定义了一个同态从多对数复形定义的Goncharov的扩展群的混合泰特动机定义的Bloch和Kriz。这张地图的存在由Beilinson-Deligne证明。Beilinson和Deligne给出了与KP 1猜想和Beilinson-Soule猜想等价的一些猜想。在不假设这些猜想的情况下，利用棒结构和恢复原理构造了同态。(2)证明了微分分次Hopf代数的余模范畴与微分分次代数上的微分分次复形的余模范畴之间的微分分次范畴等价性。利用这个等价性，证明了由Deligne代数构造的Hopf余代数对混合Tate Hodge结构的变差范畴进行了分类。(3)本文给出了正特征代数簇的pro-p完备化的刻划，它将Tannakian范畴划分为 ...更多信息 f用Bar结构表示的Fp局部系统。为了得到相应的Hopf代数上的乘积结构，我们引入了同伦混洗乘积。通过这个shuffle积，我们得到了一个类群元素的概念，它给出了pro-p完备化的描述。(4)本文定义了高阶亏格超椭圆曲线的算术-几何平均数，它是Gauss算术几何平均数的推广，并利用Gauss公式证明了它们与超椭圆曲线周期的某些行列式相一致。我们表明，这也等于一个时期的某些卡-丘品种。(5)本文构造了一类Jacobian簇与Calabi-Yau簇之间的代数对应，这两类簇是由在亏格为三的CaylayOctad的投射对偶上分支的三维投射空间的双覆盖所得到的.此外，通过考察Hodge结构的无穷小变差，证明了Calabi-Yau簇的三阶上同调一般不是外积。少