Problems in Complex Geometry

AbstractAward：DMS-9971462首席研究员：Terrence J. Napier一个复流形X称为全纯凸的，如果对于X中没有极限点的无限子集S，存在X上的非全纯函数f在S上无界。 主要研究者计划研究射影簇覆盖的全纯凸性。 他还计划工作的问题有关莱夫谢茨型定理和结构完整的卡勒流形已部分动机的研究覆盖空间。 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（或更一般的复空间）区域上的全纯函数是微分学中真实的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一个（复）空间的几何结构如何与它的全纯结构相关。 这个问题已经激发了大量的工作在数学和本身的动机工作在许多领域（例如，物理）Kahler流形是空间中的（无穷小）全纯结构与（无穷小）几何结构以自然的方式连接。 全纯凸性是几何凸性的一个类似物（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）。 对于非紧复流形，全纯凸性是一个强条件，它揭示了许多关于全纯结构的信息，这一点已经被人们认识了40多年。 因此，一个自然的问题，在更广泛的背景下，上述问题是什么时候是一个Kahler流形全纯凸？

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?