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Moduli space of bounded complete Reinhardt domains and complex plateau problem.

机译:有界完整Reinhardt域的模量空间和复杂的高原问题。

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摘要

One of the most fundamental problems in complex geometry is to determine when two bounded domains in Cn are biholomorphically equivalent. Even for complete Reinhardt domains, this fundamental problem remained unsolved for many years. Using the Bergmann function theory, we construct an infinite family of numerical invariants from the Bergman functions for complete Reinhardt domains in Cn . These infinite family of numerical invariants are actually a complete set of invariants if the domains are pseudoconvex with C 1 boundaries. For bounded complete Reinhardt domains with real analytic boundaries, the complete set of numerical invariants can be reduced dramatically although the set is still infinite. We shall also discuss the role of the Hilbert 14th problem in the construction of numerical biholomorphic invariants of complete Reinhardt domains in Cn .;Moreover, for n = 2, we can construct an infinite family of numerical invariants from the Bergman functions for such domains in An-variety {(x, y, z) ∈ C3 : xy = zn +1}. These infinite family of numerical invariants are actually a complete set of invariants for either the set of all bounded strictly pseudoconvex complete Reinhardt domain in An variety or the set of all bounded pseudoconvex complete Reinhardt domains with real analytic boundaries in An variety. In particular the moduli space of these domains in An variety is constructed explicitly as the image of this complete family of numerical invariants. It is well known that An variety is the quotient of cyclic group of order n + 1 on C2 . We prove that the moduli space of bounded complete Reinhardt domains in An variety coincides with the moduli space of the corresponding bounded complete Reinhardt domains in C2 .;Another natural fundamental questions of complex geometry is to study the boundaries of complex varieties. For example, the famous classical complex Plateau problem asks which odd dimensional real sub-manifolds of CN are boundaries of complex sub-manifolds in CN . Let X be a compact connected strictly pseudoconvex CR manifold of real dimension 2n - 1 in Cn+1 . For n ≥ 3, Yau used Kohn-Rossi cohomology groups to solve the classical complex Plateau problem in 1981. For n = 2, the problem has remained unsolved for over a quarter of a century. In this paper, we introduce a new CR invariant of X to solve this problem completely.
机译:复杂几何中最基本的问题之一是确定Cn中的两个有界域何时是双全纯的。即使对于完整的莱因哈特(Reinhardt)领域,这一基本问题多年来仍未解决。使用Bergmann函数理论,我们从Bergman函数构造了Cn中完整Reinhardt域的无穷数值不变量族。如果这些域是具有C 1边界的伪凸,则这些无限的数值不变量族实际上是一个完整的不变量集。对于具有真实解析边界的有界完整Reinhardt域,尽管数值不变量的完整集合仍然是无穷大的,但它可以大大减少。我们还将讨论希尔伯特(14)问题在构造Cn中完整Reinhardt域的数值双全纯不变量的过程中的作用;此外,对于n = 2,我们可以根据这样的域的Bergman函数构造一个无限的数值不变量族多元{(x,y,z)∈C3:xy = zn +1}。这些无穷个数值不变式族实际上是A变体中所有有界严格伪凸完整Reinhardt域的集合或A变体中具有实分析边界的所有有界伪凸完整Reinhardt域的集合的不变量的完整集合。尤其是,在A变体中,这些域的模空间被明确构造为此完整的数字不变量族的图像。众所周知,变体是C2上n阶1的环状基团的商。我们证明了A中有界完整Reinhardt域的模空间与C2中相应有界完整Reinhardt域的模空间是一致的;复杂几何的另一个自然的基本问题是研究复杂品种的边界。例如,著名的古典复杂高原问题询问CN的哪些奇数维实子流形是CN中复杂子流形的边界。令X为Cn + 1中实数为2n-1的紧连通严格伪凸CR流形。对于n≥3,Yau在1981年使用Kohn-Rossi同调学组解决了经典的复杂高原问题。对于n = 2,该问题在25年的时间里一直没有解决。在本文中,我们引入了一个新的X的CR不变量来完全解决该问题。

著录项

  • 作者

    Du, Rong.;

  • 作者单位

    University of Illinois at Chicago.;

  • 授予单位 University of Illinois at Chicago.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2009
  • 页码 134 p.
  • 总页数 134
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 遥感技术;
  • 关键词

  • 入库时间 2022-08-17 11:37:50

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