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

伪厄米对称空间被某些算术群所度量，常常被实现为各种类型的极化紧致Kahler流形的模空间，并由此阐明了它们的结构.从这一观点出发，我们得到了以下结果. (1)对于一般复环面，我们分类了它的有理自同态环的可能类型，以及相应的伪埃尔米特对称空间。(2)我们得到了二维复环面的有限自同构群的完全分类，它反映了商空间奇异性的结构。特别地，我们得到了复环面的模空间以固定抽象群为自同构群的明确描述，作为一个伪埃尔米特空间的商的算术群。(3)我们发现（2）中的空间与某些根格密切相关. (1)作为紧致Kahler辛流形的一般结构定理，我们得到了Lefschetz-Hodge分解定理的一个类比，以及第二上同调群上自然2n-形式的一些显著性质. (2)作为（1）的应用，我们已经证明了suffh流形的局部模空间的光滑性;这表明它们的模空间本质上是实现为IV. 3型埃尔米特对称空间的开子集。在一般情况下，使用的概念极值度量由于卡拉比，我们已经介绍了类似的概念，在代数几何的稳定性，并获得了一个全新的制定的模理论。这一理论在将来似乎还有许多有待发展的地方。

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.