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.

某些算术群的伪热对称汁液的伪造通常被视为各种极化紧凑型卡勒歧管的模量空间，并且通过这个事实，它们的结构被澄清了。从这个角度来看，我们获得了以下结果1。（1）对于一般的复合物托里，我们对其有理内态环的可能类型和相应的伪 - 温米特对称空间进行了分类。（2）我们已经获得了第二个维数的复杂托里的有限自动形态组的完整分类，这反映了商空间的奇异性结构。特别是，我们已经获得了与固定抽象组作为自动形态组的复杂Tori的模量空间的明确描述，作为算术组的伪空间空间的商。（3）我们发现（2）中的空间与某些根晶格密切相关。2。（1）作为紧凑型Kahler符号歧管的一般结构定理，我们获得了Lefschetz-Hodge分解定理的类比，并且在第二个共同体学组上具有自然2N形式的某些显着特性。（2）作为（1）的应用，我们显示了Sufh歧管的局部模量空间的平滑度；这表明他们的模量空间基本上被实现为IV.3型遗传学对称空间的一个开放子集。通常，使用Calabi引起的极端指标的概念，我们以与代数几何形状相比引入了稳定概念，并获得了模量理论的全新表述。该理论的未来似乎有很多发展。