Period integral of algebraic varieties

(1)定义了Goncharov定义的复数复合体到Bloch和Kriz定义的混合Tate动机的可拓群的同态。这张地图的存在是由贝林森-德列涅推测出来的。Beilinson和Deligne假设了一些与KP1猜想和Beilinson- soule猜想等价的猜想。我们在不假设这些猜想的情况下，利用棒状结构和恢复原理构造了同态。(2)证明了微分梯度Hopf代数的微模范畴与微分梯度复范畴上的微分梯度复范畴的微分梯度范畴等价。利用这一等价，我们证明了由Deligne代数构造的Hopf协代数对混合Tate Hodge结构的变分范畴进行了分类。(3)我们给出了正特征代数变体的pro-p补全的描述，将Tannakian范畴从Bar结构上划分为…More of Fp局部系统。为了得到相应Hopf代数上的积结构，我们引入了一个同伦洗牌积。通过这个洗牌积，我们得到了类群元素的概念，并给出了对pro-p补全的描述。(4)定义了高属超椭圆曲线的算术几何平均值，这是高斯算术几何平均值的推广，并利用Thomae公式证明了它们与超椭圆曲线周期的某些行列式重合。我们表明，这也等于某些葫芦品种的一个时期。(5)构造了由三维射影空间的双重覆盖得到的Jacobian变量与Calabi-Yau变量之间的代数对应关系，该复盖是在三格曲线的Caylay八元的射影对偶上分支的。更重要的是，通过观察霍奇结构的无限小变化，Calabi-Yau品种的第三次上同调一般不是外积。少

(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