Studies on meromorphic convexity in Stein spaces

Stein空间中的亚纯凸性研究

• 批准号: 18540188
• 负责人: ABE Makoto
• 金额: $ 1.39万
• 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: 2006
• 资助国家: 日本
• 起止时间: 2006 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540188/
• 关键词: complex space; Stein space; Stein manifold; meromorphic convexity; rational convexity; meromorphic approximation theorem; Cartier divisor; Picard group

Connected to the meromorphic convexity in Stein spaces, Makoto Abe obtained the following results.(1) It is proved that there exists an open set D of C^n which is not polynomially convex and satisfies the strong disk property in C^n if n≧2. This also gives a negative answer to the problem of Bremermann(1958).(2) By considering the local Steinness for an open set of possibly non-reduced Cohen-Macaulay Stein space, it is proved that the Steinness of an open set D of a Stein manifold X of dimension n satisfying the condition that dim H^k(D, O)< +∞ (2≦k ≦n-1) by the surjectivity of the canonical homomorphism Div(D)→Pic(D).(3) By giving examples, it is proved that a rationally convex open set D of C^n, where n≧2, does not satisfy in general the strong meromorphic approximation property in C^n. It is also proved that every open set D of a reduced Stein space X of dimension 1 satisfies the strong meromorphic approximation property in X.(4) A generalized meromorphic approximation theorem in a reduced Stein space X with respect to a subsemigroup G of the Picard group Pic(X)is obtained, which includes both two different meromorphic approximation theorems due to Hirschowitz(1971) and due to Abe (2005).Tadashi Shima(with Mikio Furushima and Yasuhiro Ohshima) obtained the following result.(5) Let G be a small finite subgroup of general linear group GL(2, C). Then the classification of the dual graphs of the boundaries in the minimal normal analytic compactifications of C^2/G is completed, which is a continuation of the earlier works due to Abe-Furushima-Yamasaki (2000) and Abe-Furushima-Shima (2004).

与Stein空间中的亚纯凸性相联系，Makoto Abe得到了以下结果。(1)证明了C ^n中存在一个非多项式凸的开集D，当n ≥ 2时，该开集D满足强圆盘性质。这也对Bremermann（1958）的问题给出了否定的答案。(2)通过考虑可能非约Cohen-Macaulay Stein空间开集的局部Steinness，利用标准同态Div（D）→Pic（D）的满射性，证明了n维Stein流形X的开集D的Steinness满足dim H^k（D，O）< +∞（2 <$k <$n-1）的条件. (3)通过举例证明了C^n中的有理凸开集D（n ≠ 2）一般不满足C^n中的强亚纯逼近性质.证明了1维约化Stein空间X的每个开集D都满足X中的强亚纯逼近性质。(4)得到了约化Stein空间X中关于Picard群Pic（X）的子半群G的一个广义亚纯逼近定理，它包含了Hirschowitz（1971）和Abe（2005）的两个不同的亚纯逼近定理。(5)设G是一般线性群GL（2，C）的有限小子群.然后对C^2/G的极小正规解析紧化中的边界对偶图进行了分类，这是Abe-Furushima-Yamasaki（2000）和Abe-Furushima-Shima（2004）的工作的继续.

项目成果

On open sets with the strong meromorphic approximation property(in Japanese)

具有强亚纯近似性质的开集（日语）

• 发表时间: 2007
• 作者: Makoto;Abe
• 通讯作者: Abe

Open sets satisfying the strong meromorphic approximation property

满足强亚纯近似性质的开集

• 发表时间: 2006
• 期刊: Toyama Math. J. 28(accepted)
• 作者: 水谷祐介;内田裕久;Baryshev Alexander;井上光輝;Makoto Abe
• 通讯作者: Makoto Abe

Stein多様体の領域上の正則直線束と因子

斯坦因流形域上的正则线丛和因子

• 发表时间: 2007
• 作者: 鍋島陽一;今井真一郎;名和田 新;阿部 誠
• 通讯作者: 阿部 誠

Meromorphic approximation theorems in a Stein space(in Japanese)

斯坦因空间中的亚纯逼近定理（日语）

• 发表时间: 2007
• 作者: Makoto;Abe
• 通讯作者: Abe

「研究成果報告書概要(和文)」より

摘自《研究结果报告摘要（日文）》

• 发表时间: 2005
• 作者: Kawauchi;et. al.;Nishimura et al.;Dezawa et al.;Yoshizawa et al.;星野 幹雄;星野 幹雄
• 通讯作者: 星野 幹雄