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.

本研究涉及高维复杂代数几何的主题。它主要集中在复杂射影品种的双几何中的自然问题，如一般类型品种的多象图的研究，以及理解有理曲线和全纯1型如何影响这些品种的几何。要研究的一些主要问题是：对于m的什么值是各种一般类型双民族的第m个多典型映射？具有轻微奇点的各种分辨率的纤维是否总是合理地链式连接？在一般类型的光滑投影变体上，全纯1型是否总是有非空零集？复杂代数曲面的分类是二十世纪初由意大利学派提出的。人们早就了解了它的主要特征。在高维复杂代数几何中，人们希望将理论曲面扩展到更大或更大的三维。例如，最小模型程序旨在表明，在曲面的情况下，任何变化都被轨道上等效于最小模型的有理曲线所覆盖。最小模型程序仅在3维上是完整的，但它激发了代数几何的许多重要进展。这个项目希望回答在这种情况下自然产生的一些问题和猜想。