Studies on meromorphic convexity in Stein spaces

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.

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).