Mathematical Sciences: Holomorphic Mappings

摘要建议：DMS-9622594 PI：Pinchuk拟议的研究将从三个方向进行。第一个是研究这样一个猜想：在复n-空间中，具有实解析边界的有界域之间的任何真全纯映射都扩张到它们闭包的邻域之间的真全纯映射。在维度1中，这是一个被称为Schwarz反射原理的基本结果。对于N1，该猜想仅在特殊情况下被证明。Pinchuk将使用基于Segre变种技术的几何方法，它是研究具有真实解析边界的区域中各种问题的强大的新工具。该提案的第一部分的目标是发展高维的几何反射原理，并从总体上证明上述猜想。第二个方向是应用Segre变分技巧和其他几何方法来研究全纯映射的刚性现象。也就是说，Pinchuk计划描述具有非紧自同构群的域。他还将讨论相关问题，即在相当一般的情况下，真全纯自映射是双全纯的。第三个方向是关于雅可比猜想的所谓Abhyankar方程(AEs)的研究。JC(在复数情况下，这是主要的)声称任何局部必然(双全纯)多项式映射是全局必然的。JC是数学中最耐人寻味的问题之一。自1939年以来，许多数学家(包括非常著名的数学家)对它进行了深入的研究。已经发表了一些部分结果和错误的证据。大多数专家相信，现在仍然相信JC是真实的。因此，最近一个由Pinchuk(使用AES)构建的JC真实版本的反例令人惊讶。JC在不同的数学领域有相同的公式。其中，它的解(正的或负的)将对反演公式的有限性、某些微分方程解的整体存在性以及代数集的奇性产生影响。这一建议的其他问题是自然的、重要的，被称为几个复杂变量中的困难问题。它们的解将成为全纯映射理论的重要组成部分。

ABSTRACT Proposal: DMS-9622594 PI: Pinchuk The proposed research will be carried out in three directions. The first is the study of the conjecture that in complex n-space any proper holomorphic mapping between bounded domains with real analytic boundaries extends to a proper holomorphic mapping between neighborhoods of their closures. In dimension 1 this is a fundamental result known as the Schwarz Reflection Principle. For n1 the conjecture has been proved only in special cases. Pinchuk will use a geometric approach, which is based on the technique of Segre varieties, and is a powerful new tool to study various problems in domains with real analytic boundaries. The goal of the first part of the proposal is to develop the geometric reflection principle for higher dimensions and prove the conjecture above in general. The second direction is to apply the technique of Segre varieties together with other geometric methods to study rigidity phenomena of holomorphic mappings. Namely, Pinchuk plans to describe domains with noncompact automorphism groups. He will also attack the related problem that in rather general situations proper holomorphic self-mappings are biholomorphic. The third direction is the investigation of the so called Abhyankar Equations (AEs) with respect to the Jacobian Conjecture (JC). The JC (in the complex case, which is the principal one) claims that any locally inevitable (biholomorphic) polynomial mapping is globally inevitable. The JC is one of the most intriguing problems in mathematics. It has been intensively studied by many mathematicians (including very famous ones) since 1939. A number of partial results as well as faulty proofs has been published. Most of the experts believed and still believe that the JC is true. Therefore a recent counterexample to the real version of the JC, which was constructed by Pinchuk (using the Aes) came as a surprise. The JC has equivalent formulations in different areas of mathematics. Among others, its resolution ( positive or negative) will have implications for the finiteness of inversion formulas, for the existence of global solutions of certain differential equations, and for singularities of algebraic sets. The other problems of this proposal are natural, important, and known as hard problems in several complex variables. Their solution will became the essential part of the theory of holomorphic mappings.