Birational geometry of complex projective varieties

复射影簇的双有理几何

• 批准号: 0456363
• 负责人: Christopher Hacon
• 金额: $ 10.01万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0456363&HistoricalAwards=false
• 关键词: Birational; geometry; complex; projective; varieties

The proposed research deals with topics in Higher Dimensional Complex Algebraic Geometry. It is mainly focused on natural questions inthe birational geometry of complex projective varietiessuch as the study of the pluricanonical maps of varieties of general typeand understanding how rational curves and holomorphic 1-forms influence the geometry of these varieties.Some of the main problems to be investigated are: For what values of m is the m-th pluricanonical map of a variety of general type birational?Are the fibers of a resolution of a variety with mild singularities always rationally chain connected? Is it the case that on a smooth projective variety of general type,holomorphic 1-forms always have non-empty zero set?The classification of complex algebraic surfaces was initiated by theItalian school at the beginning of the twentieth century. Its main featureshave been long understood.In Higher Dimensional Complex Algebraic Geometry, one hopes to extend the theory surfaces to dimensions grater or equal to three.The minimal model program, for example aims to show, as in the case ofsurfaces, that any variety is either covered by rational curves orbirationally equivalent to a minimal model.The minimal model program is complete only in dimension 3, but it has inspired many important advances in algebraic geometry.This project hopes to answer some of the questions and conjectures that naturally arise in this context.

拟议的研究涉及较高维代数几何形状的主题。它主要集中在复杂的射射品种的外观几何形状中的自然问题上，因为研究了多种类型的多种变种的多种形式的图形，理解了理性曲线和全体形态1形的几何形式如何影响这些品种的几何。一种具有轻度奇异性的各种分辨率总是合理地连接的？是这样的情况下，在普通的一般类型上，全态1形始终具有非空的零集？复杂代数表面的分类是由Theitalian School在20世纪初开始的。长期以来，人们已经理解了它的主要特征。代数几何形状的进步。该项目希望回答在这种情况下自然出现的一些问题和猜想。