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型L(F)在每个点至多有q-1个非正本征值；如果X允许穷举函数在某个紧子集K的补集上强q-凸，则称流形X为强q-凸的(如果X可以取K为空，则X是q-完备的)。关于复空间的q-凸性也有一些众所周知的概念。如果g是X上的Hermite度量，则f是SP(g，q)类，如果L(F)对切空间中任意一点的q维复向量子空间的限制迹为正。这样的函数是强q-凸的。首席调查员和Mohan Ramachandran应用这些函数得到了关于Levi问题和完备Kaehler流形的结构的结果。他们还开发并应用了复杂空间上的类似课程。主要研究人员计划应用这些函数将Kaehler流形的一些结果推广到奇异Kaehler空间，并研究覆盖的q-凸性，特别是(拟)射影簇的覆盖空间。拟议的部分或全部工作可能涉及与迈克尔·弗拉博尼、CezarJoita或Mohan Ramachandran的合作。复空间(特别是Stein空间和射影空间)是几个复变量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，如果连接该区域中两点的任何线段完全位于该区域内)。复数空间(如射影簇的覆盖空间)的凸性与该空间的全纯函数理论和几何密切相关，复数空间上的全纯函数(如复数平面上的区域上的全纯函数)是微积分在实数行上可微函数的自然类似。因此，研究复杂空间的凸性性质是研究复杂空间的一个基本内容。此外，凸性的概念(与对称概念一样)几乎出现在数学、科学和工程的各个领域。因此，对复杂空间的凸性性质的研究进一步证明了凸性的重要性。