Problems in Complex Geometry

摘要/ abstract摘要：dms - 9971464主要研究者：Terrence J. napia复流形X是全纯凸的，如果在X上没有极限点的无限子集S，在X上存在一个在S上无界的自纯函数f。主要研究者计划研究射影簇覆盖的全纯凸性。他还计划研究有关Lefschetz型定理和完全Kahler流形的结构的问题，这些问题部分是由覆盖空间的研究引起的。他建议将奇异度量理论的最新进展应用于这些问题。复数平面（或更一般的复数空间）区域上的全纯函数是微分学中实数线上的可微函数的自然类似物。一个非常广泛、深刻和古老的问题是确定（复）空间的几何结构与它的全纯结构之间的关系。这个问题激发了大量的数学研究，也激发了许多领域（例如物理学）的研究。Kahler流形是（无穷小）全纯结构与（无穷小）几何结构以自然方式连接的空间。全纯凸性是几何凸性的类似物（例如，平面上的一个区域是几何凸的，如果连接该区域中两点的任何线段完全位于该区域内）。四十年来人们已经知道，对于非紧复流形，全纯凸性是一个很好的条件，它能说明全纯结构的许多性质。因此，在上面提到的更广泛的问题的背景下，一个自然的问题是：Kahler流形何时是全纯凸的？

AbstractAward: DMS-9971462Principal Investigator: Terrence J. NapierA complex manifold X is said to be holomorphically convex if, forevery infinite subset S without limit points in X, there is aholomorphic function f on X which is unbounded on S. Theprincipal investigator plans to work on holomorphic convexityproperties of coverings of projective varieties. He also plansto work on problems concerning Lefschetz type theorems and thestructure of complete Kahler manifolds which have been partlymotivated by the study of covering spaces. He proposes to applyrecent advances in the theory of singular metrics to theseproblems.Holomorphic functions on regions in the complex number plane (oron more general complex spaces) are the natural analogues ofdifferentiable functions on the real number line fromdifferential calculus. A very broad, deep, and old problem is todetermine how the geometric structure of a (complex) space isrelated to its holomorphic structure. This question hasmotivated a large amount of work in mathematics and has itselfbeen motivated by work in many fields (for example, physics).Kahler manifolds are spaces in which the (infinitesimal)holomorphic structure is connected with the (infinitesimal)geometric structure in a natural way. Holomorphic convexity isan analogue of geometric convexity (for example, a region in theplane is geometrically convex if any line segment connecting twopoints in the region lies entirely within the region). It hasbeen known for forty years that, for a noncompact complexmanifold, holomorphic convexity is a strong condition which tellsone a great deal about the holomorphic structure. Thus a naturalquestion in the context of the broader problem mentioned above iswhen is a Kahler manifold holomorphically convex?