Algebro-geometric studies of rational singularities and related singularities by blowing-ups

通过爆炸对有理奇点和相关奇点进行代数几何研究

• 批准号: 09640021
• 负责人: TOMARI Masataka
• 金额: $ 1.86万
• 依托单位: Kanazawa University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640021/
• 关键词: the normal graded rings; rational singularities; weighted blowing-ups; conjecture of M.Reid; the Segre product; terminal singularities; the arithemetic genus; the special log forms; ネバンリンナ理論; 単純K3特異点; 因子的ブロ-イングアップ; 正則曲線の一意性定理; 多様体のホモトピー同値; 正則

On the main theme of this project :(1)In 1997, M.Tomari found more 3 examples of simple K3 singularities which do not belong to the famous 95 classess. It was a natural continuation of studies of previous year. Tomari also found a. counter example to an analogus conjecture of M.Reid about 4-dimensional terminal singularities in terms of Newton boundary. In the both studies, the theory of filtered blowing-up by Tomari-Watanabe plays an essential role. In 1998, Tomari succeeded to prove the criterion about the rational singularities and isolated singularities about the Segre product of two normal graded rings. The criterions are natural generalizations to those for the normal graded rings in terms of Pinkham-Demazure's construction.(2)T.Hayakawa studied several partial resolutions of 3-dimensional terminal singularities by weighted blowing-ups. In particular he succeded to show a special corespondence between the set of divisorial blowing-ups with minimal discrepancy and the set of the maximal blowing-ups with "big weight". He classified the elementary contraction with the minimal discrepancy in his situation.(3)M.Takamura gave a very good estiamte about the arithemetic genus of normal two-dimensional singularities of multiplicity two in terms of the Horikawa canonical resolution. Combined with the previous result of Tomari, he obtained the complete classification of the case of p_<alpha> = 2.As related works on complex analysis :(4)K.Morita studied the special log forms which gives a-basis of higher dimensional de Rham cohomology which is related to the arrangements of hyperfurface on the complex affine space. The work is aimed to give application to integral representaion of hypergeomeric functions of several variables and a natural generalization of Aomoto-Kita's theory.

（1）1997年，M.托马里发现了3个不属于著名的95类的简单K3奇点。这是前一年研究的自然延续。托马里还发现了一个。M.Reid关于四维终端奇点的一个类比猜想的反例。在这两个研究中，Tomari-Watanabe的过滤爆破理论起着至关重要的作用。1998年，托马里成功地证明了两个正规分次环的塞格雷积的有理奇异性和孤立奇异性的判别准则。根据Pinkham-Demazure的构造，这些判别准则是正规分次环判别准则的自然推广.（2）T.Hayakawa利用加权blowing up研究了三维终端奇点的几种部分分解。特别是他成功地显示了一个特殊的correspondence之间的一套divisorial爆破最小的差异和一套最大的爆破与“大重量”。他把这种基本的收缩与他的处境中的最小差异归类。（3）M.Takamura利用Horikawa正则分解对重数为2的二维正规奇点的算术亏格给出了很好的估计。结合托马里以前的结果，他得到了p_ = 2情形的完全分类<alpha>。作为复分析的相关工作：（4）K.Morita研究了特殊的对数形式，它给出了与复仿射空间上超曲面的排列有关的高维de Rham上同调的一个基。本文的工作旨在应用于多元超几何函数的积分表示和Aomoto-Kita理论的自然推广。

项目成果

M.Morishita and T.Watanabe: "On 5-Hardy Littlewood homogeneous spaces." Intern.J Math. vol 9. 723-757 (1998)

M.Morishita 和 T.Watanabe：“关于 5-Hardy Littlewood 均质空间。”

M.Morishita and T.Watanabe: "On S-Hardy-Littlewood homogeneous spaces" Intern J.Math.vol.9. 723-757 (1998)

M.Morishita 和 T.Watanabe：“论 S-Hardy-Littlewood 齐次空间” Intern J.Math.vol.9。

A.Kodama: "A characterization of certain weakly pseudoconvex domain" Tohoku Math.J.vol 51to appear. 未定 (1999)

A.Kodama：“某些弱赝凸域的表征”Tohoku Math.J.vol 51 待定（1999）。

T.Hayakawa: "Blowing ups of 3-dimensional terminal singularities" Publ.Res.Inst.Math.Sci.Kyoto Univ.to appear.

T.Hayakawa：“3维终端奇点的爆炸”Publ.Res.Inst.Math.Sci.Kyoto Univ.出现。

K.Morita: "On the basis of twisted de Rham cohawology" Hokkaido Math. J.vol.27. 567-603 (1998)

K.Morita：“基于扭曲的 de Rham cohawology”北海道数学。