Geometric Complex Analysis

在值分布理论中，建立了多元函数域的Nevanlinna-Cartan理论，作为应用，给出了有限性定理。在半交换簇的情形下，Noguchi，Yamanoi，Winkelmann证明了第二主要定理，他们还证明了整条曲线的分布类似于整点的分布，并得到了它们的维数估计。关于全纯映射的唯一性定理，Aihara，Fujimoto，Shirosaki等人得到了一些结果。在CR几何中，Miyajima完成了Kuranishi关于复结构变形的程序，Hirachi解决了Bergman核在强伪凸边界附近的渐近展开的猜想，Komatsu，Kamimoto得到了新的CR不变量。Ohsawa证明了P^2中新的整除、L^2扩张定理和真实的解析Levi-平坦超曲面的不存在性。Kazama，Abe，Umeno在复李群的函数论性质，特别是复拟环面上得到了新的结果，Shimizu在管域的全纯等价问题上得到了新的结果，中野关于弱1-完备流形的猜想由Takayama和Abe应用于半交换簇的Lefschetz定理而解决。Mabuchi认为Bando-Calabi-Futaki特征是半稳定性的障碍。Yoshikawa在解析挠率的自同构性质方面取得了一些进展，Tsuji在解析Zapraki分解的应用方面取得了一些进展，其他方面，上田、西村在复微分系统方面取得了新的结果，寺田、加藤在超几何函数方面取得了新的结果，托马里、大久间、Tsuji在奇点及其变形方面取得了新的结果。目前的研究项目由Noguchi指导。

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.