Geometric Complex Analysis

几何复分析

• 批准号: 09304014
• 负责人: NOGUCHI Junjiro
• 金额: $ 15.87万
• 依托单位: University of Tokyo (1998-2000)Tokyo Institute of Technology (1997)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A).
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09304014/
• 关键词: several complex variables; geometric complex analysis; Kobayahsi hyperbolic manifold; value distribution; invariant metric; moduli space; pseudoconvex domain; Teichmuller space; モジュライ理論; タイヒミューラー理論

In value distribution theory, the Nevanlinna-Cartan theory was established for function fields of several variables, which yields finitness theorems as applications. In the case of semi-abelian variety, the Second Main Theorem was proved by Noguchi, Yamanoi, Winkelmann, who also showed that the distribution of entire curves is analogous to that of integral points, and obtained their dimension estimates. On uniqeness theorem of holomorphic mappings, some results were obtained by Aihara, Fujimoto, Shirosaki. New examples of hyperbolic projective hypersurfaces were constructed by Shirosaki, Fujimoto.In CR geometry, Kuranishi's program on deformation of complex strutures was accomplished by Miyajima, Fefferman's conjecture on the asymptotic expansion of the Bergman kernel near the strongly pseudo-convex boundary was settled by Hirachi, and new CR invariants were obtained by Komatsu, Kamimoto. Ohsawa proved new divison, L^2 extension theorems, and the non-existence of real analytic Levi-flat hypersurfaces in P^2.New results on function theoretic property of complex Lie groups, especially complex quasi-tori were obtained by Kazama, Abe, Umeno, those on the holomorphic equivalence problem of tube domains by Shimizu, and those on the characterization of ellipsoids by their boundary by Kodama.Nakano's conjecture on weakly 1-complete manifolds was solved by Takayama with application by himself and Abe to the Lefschetz theorem for semi-abelian varieties. By Mabuchi, the Bando-Calabi-Futaki character was shown to be an obstruction to the semi-stability. Some progress were made by Yoshikawa on the automorphic property of analytic torsion, and by Tsuji concerning applications of analytic Zariski decompositions.About other areas, new results on complex dinamical systems were obtained by Ueda, Nishimura, those on hypergeometric functions by Terada, Kato, and those on singularities and their deformations by Tomari, Okuma, Tsuji. The present research project was directed by Noguchi.

在值分布理论中，建立了多变量函数域的Nevanlinna-Cartan理论，并给出了应用的有限性定理。在半阿贝尔变的情况下，Noguchi, Yamanoi， Winkelmann证明了第二主定理，他们还证明了整条曲线的分布类似于积分点的分布，并得到了它们的维数估计。关于全纯映射的唯一性定理，得到了Aihara， Fujimoto， Shirosaki等人的一些结果。Shirosaki， Fujimoto构造了双曲射影超曲面的新例子。在CR几何中，Miyajima完成了Kuranishi关于复杂结构变形的规划，Hirachi完成了Fefferman关于Bergman核在强伪凸边界附近渐近展开的猜想，Komatsu， Kamimoto获得了新的CR不变量。Ohsawa证明了新的除法，L^2的可拓定理，以及P^2中实解析列维平面超曲面的不存在性。Kazama, Abe， Umeno等人获得了复李群，特别是复拟环面泛函性质的新结果，Shimizu等人获得了管域全纯等价问题的新结果，Kodama等人获得了椭球体边界表征的新结果。Takayama和Abe将Nakano关于弱1-完全流形的猜想应用于半阿贝变分的Lefschetz定理，求解了Nakano关于弱1-完全流形的猜想。Mabuchi证明了Bando-Calabi-Futaki特征是半稳定性的障碍。Yoshikawa在解析扭转的自同构性质方面取得了一些进展，Tsuji在解析Zariski分解的应用方面取得了一些进展。在其他领域，Ueda, Nishimura, Terada， Kato以及Tomari， Okuma， Tsuji在奇异点及其变形方面取得了关于复杂动力系统的新成果。目前的研究项目是由野口主持的。

项目成果

Y.Abe and P.Zappa: "Universal functions on complex general linear groups"J.Approximation Theory. 100. 221-232 (1999)

Y.Abe 和 P.Zappa：“复杂一般线性群上的通用函数”J.近似理论。

Y.Abe: "Meromorphic functions admitting an algebraic addition theorem"Osaka J.Math.. 36. 343-363 (1999)

Y.Abe：“承认代数加法定理的亚纯函数”Osaka J.Math.. 36. 343-363 (1999)

Y.Abe: "Universal functions on complex special linear groups"Proceedings of the Fourth International Conference on Difference Equations, Poznan, 1998. 1-8 (2000)

Y.Abe：“复杂特殊线性群上的通用函数”第四届国际差分方程会议论文集，波兹南，1998. 1-8 (2000)

Yoshihiro Aihara: "Unicity theorems for meromorphic mappings with deficiencies"Complex Variables. 42. 259-268 (2000)

Yoshihiro Aihara：“具有缺陷的亚纯映射的唯一性定理”复杂变量。

Yoshihiro Aihara: "Algebraic dependence of meromorphic mappings with deficiencies into algebraic varieties"Proc.ISAAC'99 Fukuoka,(eds.P.Gilbert et al.), Kluwer Academic Publishers. (in press). (2000)

Yoshihiro Aihara：“亚纯映射的代数依赖性与代数簇的缺陷”Proc.ISAAC99 Fukuoka，（eds.P.Gilbert 等人），Kluwer 学术出版社。