Period integral of algebraic varieties

(1)定义了从Goncharov定义的PolyLog复形到Bloch和Kriz定义的混合Tate动机扩张群的同态。这个映射的存在是由Beilinson-Deligne猜想的。Beilinson和Deligne提出了一些猜想，这些猜想与KP1猜想和Beilinson-Soule猜想等价。(2)证明了微分分次Hopf代数的余模范畴与伴随着微分分次代数的微分分次复范畴之间的微分分次范畴等价，利用这种等价关系，我们证明了由Deligne代数构成的Hopf余代数是混合Tate-Hodge结构的变分范畴；(3)给出了正特征代数簇的Prop完备性的刻画，从而对…上的Tannakian范畴进行了分类在酒吧建设方面，更多的是FP当地系统。为了得到相应Hopf代数上的乘积结构，我们引入了一个同伦Shuffle积。(4)定义了高亏格超椭圆曲线的算术几何平均，它是Gauss算术几何平均的推广，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。我们证明了这也等于某些Calabi-Yau簇的周期。(5)我们构造了Jacobian簇和Calabi-Yau簇之间的某种代数对应，这些簇是由在亏格三次曲线的Caylay八面体的射影对偶处分枝的三维射影空间的二重覆盖得到的。此外，通过观察Hodge结构的无限小变化，Calabi-Yau变种的第三个上同调一般不是外积。较少

(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