The structure and classification of complex analytic compactifications of C^n with the second Betti number equal to one

第二个Betti数等于1的C^n的复解析紧化的结构和分类

• 批准号: 13640082
• 负责人: FURUSHIMA Mikio
• 金额: $ 2.18万
• 依托单位: KUMAMOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640082/
• 关键词: コンパクト化; del Pezzo曲面; Moishezon; ファノ多様体; モイシェゾン多様体

We investigate mainly(1) the Fano compactifications of C^3 with hypersurface terminal singularities and with second Betti number equal to one and(2) the structure of the non-projective compactifications (X,Y) of C^3 with second Betti number equal to one.Concerning to the investigation (1) I succeeded in constructing the Fano compactifications of C^3 with hypersurface terminal singularities and second Betti number equal to one, which are essentially new.Concerning to the investigation (2) it is easy to see that the canonical divisor can be written as follow: K_X=-rY (r=1,2). When Y is nef, the structure of (X,Y) is completely determined by myself. Thus the problem is the cases where Y is not-nef. Unfortunately the structure is not known well in this case. However I can obtain some partial results. For example, the boundary Y is birational equivalent to a rational surface or a ruled surface.Furthermore, we find that some technique developed in the study of compactifications of C^3 can be applied to the classification of the non-normal del Pezzo surfaces, and I succeeded in its classification. Moreover I prove that -3K_S is always very ample, where we denote by -K_S the anti-canonical divisor of the non-normal del Pezzo surface S. This result is an affirmative answer to the question by Miyanishi. The paper is accepted in Math. Nachrichten (August, 2002).

主要研究(1)C^3的二次Betti数为1且具有超曲面端点奇点时的Fano紧化；(2)C^3的二次Betti数为1时的非射影紧化（X，Y）的结构。关于研究(1)，我成功地构造了C^3具有超曲面端点奇点和第二Betti数等于1的Fano紧化，这是本质上新的。关于研究(2)，很容易看出正则除数可以写成：K_X=-rY （r=1,2）。当Y为nef时，（X，Y）的结构完全由我自己决定。因此问题是Y不是nef的情况。不幸的是，在这种情况下，结构不是很清楚。然而，我可以得到一些部分的结果。例如，边界Y等价于有理曲面或直纹曲面。此外，我们发现在C^3紧化研究中发展的一些技术可以应用于非正态del Pezzo曲面的分类，并成功地对其进行了分类。此外，我证明了-3K_S总是非常充足的，其中我们用-K_S表示非正规del Pezzo曲面s的反正则因子，这一结果是对Miyanishi问题的肯定回答。这篇论文在数学领域被接受。Nachrichten（2002年8月）。

项目成果

M.Abe, M.Furushima: "On non-normal del Pezzo surfaces"Mathematiche Nachrichten. (近刊). (2003)

M.Abe、M.Furushima：“关于非正态 del Pezzo 曲面”Mathematice Nachrichten（即将出版）。

M.Misawa: "Local Holder regularity of gradients for evolutional p-Laplacian systems"Annali di Matematica. 181. 389-405 (2002)

M.Misawa：“演化 p-拉普拉斯系统的梯度局部保持规律”Annali di Matematica。