Algebraic-geometrical and arithmetical study of a quotient space of a Riemannian symmetric space by an arithmetic group

通过算术群对黎曼对称空间的商空间进行代数几何和算术研究

基本信息

批准号：
60540038
负责人：
FUJIKI Akira
金额：
$ 1.28万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1985
资助国家：
日本
起止时间：
1985 至 1986
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-60540038/
关键词：
Hermitian symmetric space arithmetic subgroup moduli space 複素トーラス

项目摘要

Quotinets of pseudo-hermitian symmetric sapces by certain arithmetic groups are realized often as a moduli space of various types of polarized compact Kahler manifolds, and through this fact their structures are clarified. From this point of view we have obtained the following results.1. (1) For a general complex tori, we have classifieid the possible types of its rational endomorphism rings, and the corresponding pseudo-hermitian symmetric space. (2) We have obtained the complete classification of finite automorphism groups of complex tori of dimension two, which reflects the structure of the singularity of our quotient space. Especially, we have obtained explicit description of the moduli space of complex tori with fixed abstract group as automorphism group, as a quotient of a pseudo-hermitian space by arithmetic group. (3) We have found that the spaces in (2) are closely related with certain root lattices.2. (1) As a general structure theorem for compact Kahler symplectic manifolds we have obtained an analogy of Lefschetz-Hodge decomposition theorem, and some remarkable property of a natural 2n-form on the second cohomology group. (2) As an application of (1) we have shown the smoothness of the local moduli space of sufh manifolds; this shows that their moduli space is essentially realized as an open subset of hermitian symmetric space of type IV.3. In general, using the notion of extremal metrics due to Calabi, we have introduced the notion of stability in analogy with that in algebraic geometry, and obtained an entirely new formulation of the moduli theory. There seems much to be developped in the future in this theory.

某些算术群的伪厄米对称空间的引文通常被实现为各种类型的极化紧卡勒流形的模空间，并且通过这个事实它们的结构被澄清。从这个角度出发，我们得到了以下结果： 1． (1)对于一般复环面，我们对其有理自同态环的可能类型以及相应的伪厄米对称空间进行了分类。 (2)我们得到了二维复环有限自同构群的完全分类，它反映了商空间的奇异性结构。特别是，我们获得了具有固定抽象群作为自同构群的复圆环模空间的显式描述，作为伪厄米空间与算术群的商。 (3)我们发现(2)中的空间与某些根格密切相关。2． (1) 作为紧卡勒辛流形的一般结构定理，我们得到了 Lefschetz-Hodge 分解定理的类比，以及第二上同调群上自然 2n 型的一些显着性质。 (2) 作为(1)的应用，我们证明了sufh流形的局部模空间的光滑性；这表明它们的模空间本质上被实现为 IV.3 型厄米对称空间的开子集。总的来说，利用卡拉比提出的极值度量概念，我们引入了与代数几何中类似的稳定性概念，并获得了模理论的全新表述。这一理论在未来似乎还有很多需要发展的地方。