多复变中与群作用相关问题的研究

This project is mainly to study the Stein manifolds with group actions as well as the properties and structures of the holomorphic mappings between them. It is an important view to study several complex variables by using the theory of group actions. Many famous researchers have been paying close attention to this direction. The applicants of this project have been doing research on this direction and have got some experiences and results which have been published on or accepted by Math.Z., Trans. AMS., Comm.A.G., J.M.P.(see references [11],[15],[35],[36]).. Several problems as follows will be discussed on the basis of our previous work by using the theories and tools of group actions, combining with the theories and methods in several complex variables:. To study the orbit connectedness and orbit convexity of invariant domains and their roles in the analytic continuation problems；To discuss the special geometric and analytic properties of the boundary of invariant domains, and their applications in studying of proper holomorphic mappings between invariant domains. To study the minimum principle for plurisubharmonic functions and explore the relationship between plurisubharmonic functions, Bergman kernels and group actions. To explore the generalization of the theory of vertex operator algebras to Riemann surfaces with higher genus by using of the Krichever-Novikov basis.

本项目着重研究带有群作用的Stein流形及其间的全纯映照的性质与结构。利用群作用一般理论研究多复变相关问题是一个重要视角，受到国内外知名学者的密切关注。本项目申请人近几年在该方向进行探索，有一定的工作积累，发表或接受发表在Math.Z., Trans.AMS., Comm.A.G., J.M.P.等期刊上（见参考文献[11][15][35][36]）。. 本项目拟在前期的工作基础上，利用群作用的理论和工具，结合多复变的理论和方法考虑以下问题：探讨群作用不变区域的轨道连通性及轨道凸性在多复变解析延拓问题中的作用；探讨不变区域边界具有的特殊几何、解析性质，及其在研究紧与非紧李群不变区域间的逆紧全纯映照中的运用；研讨多次调和函数的极小原理，多次调和函数、Bergman核与群作用的关系；探索利用黎曼面上的Krichever-Novikov基将顶点算子代数理论推广到高亏格黎曼面上。

利用群作用一般理论研究多复变相关问题是一个重要视角，已受到国内外知名学者的密切关注。本项目着重研究带有群作用的Stein流形及其间的全纯映照的性质与结构。本项目也研究多次调和函数、逆紧全纯映照等多复变中的核心研究对象。本项目将多位著名数学家的重要工作做了统一并拓展到更一般的框架、得到了新的结果。