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STUDY ON THE COMPLEX AFFINE SPACE CィイD1NィエD1 AND ITS COMPACTIFICATION

复仿射空间C D1N D1及其紧化研究

基本信息

批准号：
10640026
负责人：
FURUSHIMA Mikio
金额：
$ 1.92万
依托单位：
KUMAMOTO UNIVERSITY (1999)Hiroshima University (1998)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

项目摘要

We investigated projective compactifications or non-projective Moishezon compactifications of CィイD13ィエD1 and the classification of minimal normal compactifications of CィイD12ィエD1/G, where G is a small finite subgroup of the general linear group GL(2,C), and we obtained several new results. We will state as follows. There are six types of projective compactifications of CィイD13ィエD1 with second Betti number equal to one. This was obtained by Furushima before. In this research, we gave a concrete construction of these six compactifications of C3 from the well-known compactifications (PィイD13ィエD1,PィイD12ィエD1), that is, we gave an explicit birational map of PィイD13ィエD1 to X which is biregular on CィイD13ィエD1-part. This finishes the projective classifications of such compactifications of CィイD13ィエD1.Next, we also studied the structure of the non-projective compactifications (X,Y) of CィイD13ィエD1 with second Betti number equal to one. In this case, it is easy to see that the canonical divisor can be written as follow: KX=-rY (r>0 is an integer). In this research, we can show that the integer r is equal to one or two and that there are many new examples of such non-projective compactifications of CィイD13ィエD1.Furthermore, we find that some technique developed in the study of compactifications of CィイD13ィエD1 can be applied to the classification of the minimal normal compactifications of CィイD12ィエD1/G, then we succeeded in its classification.

我们研究了CィイD13ィエD_1的射影紧化或非射影Moishezon紧化以及C_ィイD_(12)ィエD_1/G的极小正规紧化的分类，其中G是一般线性群GL(2，C)的一个小的有限子群，得到了几个新的结果。我们将声明如下。C-ィイ-D13-ィエ-D-1有六种射影紧化，其第二Betti数等于1。这是古岛之前获得的。在这项研究中，我们从著名的紧化(PィイD13ィエD_1，PィイD_12ィエD_1)给出了C_3的这六个紧化的具体构造，即给出了P_ィイD_(13)ィエD_1到X的显式双正则映射C_ィイD_(13)ィエD_1-部分。其次，我们还研究了CィイD13ィエD1的非射影紧化(X，Y)的结构，其中第二个Betti数等于1。在这种情况下，很容易看到标准除数可以写成：kx=-ry(r&gt；0是一个整数)。在这项研究中，我们可以证明整数r等于1或2，并且CィイD13ィエD1的这种非射影紧化有许多新的例子。此外，我们发现在CィイD13ィエD 1的紧化研究中发展的一些技巧可以应用于CィイD12ィエD 1/G的极小正规紧化的分类，并且我们成功地分类了它。