Complex analysis of Bergman spaces and α-cohomology

Bergman 空间和 α-上同调的复分析

• 批准号: 10440041
• 负责人: OHSAWA Takeo
• 金额: $ 3.33万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10440041/
• 关键词: Bergman Kernel; LィイD12ィエD1 holomorphic function; Pseudoconvex domain; Bergman metric; Complexification of gid; Extension theorem; Division theorem; Bergman計算; ^∂_-方程式; 皆既関数; 超擬凸; Bergman空間; L^2拡張定理

Bergman metric of bounded balanced pseudoconvex domains in CィイD1nィエD1 was studied. A quantitative result implying the completeness first due to Jarnicki-Pflug was obtained. Singular fiber metric was used to obtain as estimate for the Bergman kernel function on pseudoconvex domains in PィイD1nィエD1. A new phenomenon encountere was that locally hyperconvex domains in PィイD1nィエD1 are not necessarily globally hyperconvex. As for the domains with smooth boundary existence of bounded p.s.h. functions was proved. In case the boundary is real analytic, it was proved that such a pseudoconvex domain must be strictly pseudoconvex at some boundary point. More recently, a generalization of an LィイD12ィエD1 extension theorem to complex manifolds was obtained, which has, as a corollary, a famous LィイD12ィエD1 division theorem of Skoda.

研究了C_n中有界平衡伪凸域的Bergman度量.得到了一个定量的结果，暗示了Jarnicki-Pflug的完全性。利用奇异纤维度量得到了P_n中伪凸域上Bergman核函数的估计.一个新的现象是，局部超凸域在P_n中不一定是全局超凸的。对于具有光滑边界的区域，存在有界p.s.h.功能得到证实。当边界是真实的解析边界时，证明了这样的拟凸区域在边界点处一定是严格拟凸的。最近，一个L D12 D1扩张定理被推广到复流形上，作为一个推论，它有一个著名的Skoda的L D12 D1除法定理。

项目成果

K. Diederich and T. Ohsawa: "On pseudoconvex domains in PィイD1nィエD1"Tokyo J. Math. 21. 353-358 (1998)

K. Diederich 和 T. Ohsawa：“论 PiiD1nieD1 中的伪凸域”Tokyo J. Math. 21. 353-358 (1998)

大沢健夫: "多変数複素解析" 岩波書店, 119 (1998)

大泽武夫：“多元复杂分析”岩波书店，119（1998）

T.Ohsawa and N.Sibony: "Kahler identity on Levi flat manifolds"Nagoya Math.J..

T.Ohsawa 和 N.Sibony：“Levi 平流形上的卡勒恒等式”Nagoya Math.J.。

T. Ohsawa: "Pseudoconvex domains in PィイD1nィエD1 : A question on 1-convex boundary points"Proc. of the 40th Taniguchi Symp.. 239-252 (1999)

T. Ohsawa：“PiiD1nieD1 中的伪凸域：关于 1-凸边界点的问题”Proc. 第 40 届 Taniguchi Symp.. 239-252 (1999)

K. Diederich and T. Ohsawa: "On psendoconvex domains in TP^n"Tokyo J. Math.. 21. 353-358 (1998)

K. Diederich 和 T. Ohsawa：“关于 TP^n 中的伪凸域”Tokyo J. Math.. 21. 353-358 (1998)