Studies on meromorphic convexity in Stein spaces

连接Stein空间中的亚纯凸性，Makoto Abe得到了以下结果。(1)证明了存在一个C^n的开集D，它不是多项式凸的，且当n≧2时满足C^n的强盘性。这也给了Bremermann（1958）的问题一个否定的答案。(2)通过考虑可能非约简Cohen-Macaulay Stein空间的开集的局部斯坦性，证明了n维Stein流形X的开集D的斯坦性满足dim H^k(D， O)< +∞(2≦k≦n-1)的正则同态Div(D)→Pic(D)的满射性。(3)通过举例证明了C^n的合理凸开集D在n≥2时一般不满足C^n的强亚纯逼近性质。并证明了1维的约简Stein空间X的每一个开集D都满足X上的强亚纯逼近性质。(4)得到了约简Stein空间X上关于Picard群的子半群G的广义亚纯逼近定理，该定理既包括Hirschowitz（1971）的亚纯逼近定理，也包括Abe（2005）的亚纯逼近定理。岛正（与古岛美雄和大岛康弘）得到了如下结果。(5)设G是一般线性群GL（2， C）的一个有限小群。然后完成了C^2/G的最小正规解析紧化中边界对偶图的分类，这是Abe-Furushima-Yamasaki（2000）和Abe-Furushima-Shima（2004）的早期工作的延续。

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).