Complex Variables Analogues of Hilbert's Seventeenth Problem

摘要：D’angelo的项目将继续研究希尔伯特第十七问题的复变量版本（由Catlin和D’angelo提出），并将其应用于复杂几何、分析和代数。他提出了关于全纯束的Catlin-D'Angelo等长嵌入定理的推论，并将结果推广到某些退化度量。这一工作直接涉及到不同维球间全纯映射的分类问题。希尔伯特第十七题的复变版本已经证明有如下结果。假设给定复欧几里德空间上的一个多项式，它在闭合单位球上不消失。这个多项式必须是球之间的有理适当映射（减少到最低项）的分母，但是需要选择足够大的目标维度。因此，限制目标尺寸限制了球间适当映射的复杂性。Faran的工作进一步证明了这一点。D’angelo试图证明一个一般定理，为不同维单位球之间的固有有理映射的度提供一个明确的界。这将为关于不同维的CR流形之间的CR映射的更一般的猜想提供证据。大卫·希尔伯特在1900年国际数学家大会上的著名演讲中提出了一系列研究问题，这些问题对本世纪数学的发展产生了重大影响。1925年前后，马丁证明了几个实变量的多项式，其值是非负的，必须是有理函数的平方和，从而解决了希尔伯特最初的第17个问题。这个结果的自然类比对于几个复杂变量的多项式是不成立的。考虑到复变量理论在物理、工程和纯数学中的重要性，考虑希尔伯特问题在复数多项式中的其他类似形式是很自然的。近年来，德安吉洛发现，这种类似物在研究不同复杂维度的球之间的“适当映射”中起着至关重要的作用。德安吉洛和卡特林后来在复杂背景下建立了一个版本的阿廷定理。它在适当映射中的应用使人们能够理解单位球在高维中的作用，从而为研究静电、扩散和流体流动提供了一个多维工具。最近，Catlin和D'Angelo证明了一个重要的结果，通过将他们早期的定理投射到一个抽象的框架中，即所谓的全纯向量束的框架。证明给出了伯格曼核函数（分析的一部分）对代数问题的意外应用，以及对希尔伯特问题的几何重新解释。这一结果对于现代数学物理（如弦理论和超对称）中全纯向量束发挥重要作用的部分可能具有相当大的意义。D’angelo是1999年伯格曼奖的获得者，他的作品在引文中被提及。D’angelo目前的建议是对这些想法的各种扩展和应用。

Proposal: DMS-9970024Principal Investigator: John P. D'AngeloAbstract: D'Angelo's project will continue his investigation of a complex variables version of Hilbert's seventeenth problem (due to Catlin and D'Angelo) and give applications of it to complex geometry, analysis, and algebra. He proposes to derive corollaries of the Catlin-D'Angelo isometric imbedding theorem for holomorphic bundles, and to extend the result to certain degenerate metrics. This work bears directly on the problem of classifying proper holomorphic mappings between balls in different dimensions. The complex variables version of Hilbert's seventeenth problem already proved has the following consequence. Suppose one is given a polynomial on complex Euclidean space that does not vanish on the closed unit ball. This polynomial must then be the denominator of a rational proper mapping (reduced to lowest terms) between balls, but the target dimension needs to be chosen sufficiently large. Thus restricting the target dimension constrains the complexity of a proper map between balls. Work of Faran gives further evidence of this. D'Angelo seeks to prove a general theorem providing a sharp bound for the degree of a proper rational mapping between unit balls in different dimensions. This will provide evidence for a more general conjecture about CR mappings between CR manifolds of different dimensions.In his famous address to the International Congress of Mathematicians in 1900, David Hilbert posed a series of research problems that have greatly influenced the development of mathematics in this century. Artin solved Hilbert's original Seventeenth Problem around 1925 by proving that a polynomial in several real variables whose values are nonnegative must be a sum of squares of rational functions. The natural analogue of this result for polynomials of several complex variables does not hold. Given the importance of complex variable theory in physics, engineering, and pure mathematics, it is natural to consider other facsimiles of Hilbert's problem for complex polynomials. In recent years, D'Angelo discovered that such analogues play a crucial role in the study of objects known as "proper mappings" between balls in different complex dimensions. D'Angelo and Catlin later established a version of Artin's theorem in the complex setting. Its application to proper mappings enables one to understand the role of the unit sphere in higher dimensions and thus furnishes a multidimensional tool for the study of electrostatics, diffusion, and fluid flow. More recently, Catlin and D'Angelo have proved a major result that generalizes their earlier theorems by casting them in an abstract framework, the framework of so-called holomorphic vector bundles. The proof gives an unexpected application of the Bergman kernel function (a part of analysis) to a problem in algebra, as well as a geometric reinterpretation of Hilbert's problem. The result may have considerable significance for the parts of modern mathematical physics (such as string theory and supersymmetry) where holomorphic vector bundles play a large role. D'Angelo is the 1999 winner of the Bergman Prize, and this work is mentioned in the citation. D'Angelo's current proposal suggests various extensions of and applications for these ideas.