Problems in Complex Geometry

复杂几何问题

• 批准号: 0306441
• 负责人: Terrence Napier
• 金额: $ 9.12万
• 依托单位: Lehigh University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306441&HistoricalAwards=false
• 关键词: Problems; Complex; Geometry

A smooth real-valued function f on a complex manifold Xis said to be strongly q-convex if its Levi form L(f)has at most q-1 nonpositive eigenvalues at each point.The manifold X is called strongly q-convex if X admitsan exhaustion function which is strongly q-convex on thecomplement of some compact subset K (X is q-completeif one may take K to be empty). There are also well-knownnotions of q-convexity for complex spaces. If g is a Hermitianmetric on X, then f is of class SP(g,q) if the trace ofthe restriction of L(f) to any complex vector subspace ofdimension q in the tangent space at any point in Xis positive. Such functions are strongly q-convex. The principalinvestigator and Mohan Ramachandran have applied such functions toobtain results concerning the Levi problem and the structure ofcomplete Kaehler manifolds. They have also developed and appliedanalogous classes on complex spaces. The principal investigatorplans to apply such functions to extend some results for Kaehlermanifolds to singular Kaehler spaces and to study q-convexityproperties of coverings; in particular, covering spaces of(quasi)projective varieties. Some or all of the proposed workwill probably involve collaboration with Michael Fraboni, CezarJoita, or Mohan Ramachandran. Complex spaces (in particular, Stein spaces and projectivevarieties), are the fundamental objects of study in severalcomplex variables and algebraic geometry. The notion ofq-convexity is one of the many useful generalizationsof geometric convexity (for example, a region in the plane isgeometrically convex if any line segment connecting two pointsin the region lies entirely within the region). The convexityproperties of a complex space (for example, a covering spaceof a projective variety) are intimately connected withthe space's holomorphic function theory and geometry.Holomorphic functions on complex spaces (for example, on regionsin the complex number plane) are the natural analogues ofdifferentiable functions on the real number line from differentialcalculus. Thus the study of convexity properties isan essential element in the study of complex spaces.Moreover, notions of convexity (like those of symmetry) appear inalmost every field of mathematics, science, and engineering.Hence the study of convexity properties of complex spacesprovides further evidence of the importance of convexity.

复流形X上的光滑实值函数f称为强q-凸的，如果它的Levi形式L（f）在每一点上至多有q-1个非正特征值，流形X称为强q-凸的，如果X存在一个在某个紧子集K的补上强q-凸的耗尽函数（X是q-完备的，如果K可以取为空）.对于复空间也有一些众所周知的q-凸性概念。如果g是X上的Hermite度量，则f是SP（g，q）类的，如果L（f）在X上任意一点的切空间中对任意q维复向量子空间的限制的迹是正的.这样的函数是强q-凸的。主要研究者和Mohan Ramachandran应用这样的函数得到关于Levi问题和完备Kaehler流形的结构的结果。他们还发展和应用了复空间的类似类。主要作者计划应用这样的功能，以扩大一些结果的Kaehler流形奇异Kaehler空间和研究q-convexity性质的覆盖，特别是覆盖空间的（拟）投射品种。部分或全部的工作可能会涉及与Michael Fraboni、CezarJoita或Mohan Ramachandran的合作。 复空间（特别是Stein空间和射影空间）是几个复变和代数几何的基本研究对象。 q-凸性的概念是几何凸性的许多有用的推广之一（例如，平面上的一个区域是几何凸的，如果任何连接两个点的线段完全位于该区域内）。复空间（例如，射影簇的覆盖空间）的凸性与空间的全纯函数理论和几何学密切相关。复空间（例如，复数平面上的区域）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。 因此，凸性性质的研究是复空间研究中的一个基本要素，而且，凸性（象对称性一样）的概念几乎出现在数学、科学和工程的各个领域，因此，对复空间凸性性质的研究进一步证明了凸性的重要性。