Mathematical Sciences: Complex Manifolds and Meromorphic Mappings

这个项目继续围绕几个复杂变量理论的主题进行数学研究。这项工作包括对复杂流形的研究，代表几何观点，以及对复杂函数的值分布和连续性特性的分析研究。一个主要的焦点是所谓的Hartogs现象，即在一个区域周围（如在复环中）定义的几个变量的全纯函数作为一个全纯函数继续到内区域。将在两个方向上推广这一想法。第一个是确定一个单独的分析性意味着关节的程度，而第二个是与哈托格斯现象有关，因为它与映射到复杂流形有关。尽管许多流形反映了这一性质，但也有一些没有。突出的问题仍然是在目标流形上给出条件，以便扩展属性适用于所有到流形的映射。值分布理论关注的是定义在几个复变量空间中的全纯函数可以忽略或假设给定值的程度。这种亲和关系的测量涉及到经典Nevanlinna理论的推广，该理论是在本世纪头几十年为单变量函数而发展起来的。对于非常数函数，它们对任意值的亲和力几乎总是与其他值相同。例外表示功能的缺陷-缺陷集的测量将是本次调查中主要关注的问题。

This project continues mathematical research centering on topics in the theory of several complex variables. The work involves studies of complex manifolds, representing the geometric point of view, together with analytic investigations into value distributions and continuity properties of complex functions. One primary focus concerns what is known as the Hartogs phenomenon whereby a holomorphic function of several variables defined in a region surrounding a domain (such as in a complex annulus) continues as a holomorphic function to the inner region. Work will be done in extending the idea in two directions. The first is to determine the extent to which a separate analyticity implies joint, while the second is concerned with the Hartogs phenomenon as it relates to mapping into complex manifolds. Although many manifolds reflect this property, some do not. The outstanding problem remains one of giving conditions on the target manifolds so that the extension property holds for all mappings into the manifold. Value distribution theory is concerned with the extent to which holomorphic functions defined in the space of several complex variables can omit or assume given values. The measurement of such affinities involves generalizations of the classical Nevanlinna theory which was developed for functions of a single variable during the first decades of this century. For nonconstant functions, their affinity for any value is almost always the same as for any other. The exceptions represent the deficiencies of the function - the measure of the set of deficiencies will be of primary concern during this investigation.