Minimal Model Program in Birational Geometry

双有理几何最小模型程序

• 批准号: 0801258
• 负责人: David Morrison
• 金额: $ 24.61万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801258&HistoricalAwards=false
• 关键词: Minimal; Model; Program; Birational; Geometry

The principal investigator proposes to continue the study of the classification of Projective Varieties in Birational Geometry. The Minimal Model Program, started by Mori around the 1970's, aims to generalize the classification of projective surfaces to higher dimensional varieties. This Program was successfully carried out in the 1980's for projective three-folds. The principal investigator plans to carry out the Minimal Model Program in higher dimension, aiming to complete the classification of complex projective varieties.Moroever, the principal investigator plans to extend the Minimal Model Program to a broader class of varieties defined in positive characteristic. Although the techniques involved in this program are very different, he expect to obtain results that are as strong as in the classicalMinimal Model Program. Quite apart from its own interest, it is hoped that this study will be very useful in completing the classification of complex projective varieties. Finally, the principal investigator intends to continue his study of the Kahler-Ricci flow on a wide range of projective varieties, by translating Mori's work into an analytic language.Mathematical tools and concepts have been extensively applied in a wide range of sciences such as physics, engineering and economics. In particular, birational geometry has proven to be a very useful tool in theoretical physics, especially in string theory. Cascini's recent work has already inspired several important conferences. In particular, the Mathematical Sciences Research Institute organized a one-week workshop entitled "Hot Topics:Minimal and Canonical Models in Algebraic Geometry" to discuss the aforementioned results obtained by Cascini and his collaborators. At the same time, seminars on the same topics were organized in many departments of Mathematics in this country.

主要研究者建议继续研究双有理几何中的射影簇的分类。最小模型程序，开始由森在20世纪70年代左右，旨在推广分类的投影曲面，以更高的维品种。该计划在20世纪80年代成功地进行了投影三重。主要研究者计划在更高的维度上进行最小模型程序，旨在完成复杂投射簇的分类。此外，主要研究者计划将最小模型程序扩展到更广泛的正特征定义的簇。虽然这个程序所涉及的技术是非常不同的，他希望获得的结果是强大的，因为在classicalMinimal Model Program。除了本身的兴趣之外，我们希望这项研究对完成复射影簇的分类是非常有用的。最后，主要研究者打算通过将Mori的工作转化为分析语言，继续研究Kahler-Ricci流的各种投影变体。数学工具和概念已广泛应用于物理学、工程学和经济学等广泛的科学领域。特别是，双有理几何已被证明是理论物理中非常有用的工具，特别是在弦理论中。卡西尼最近的工作已经启发了几个重要的会议。特别是，数学科学研究所组织了一个为期一周的讲习班，题为“热门话题：最小和典型的模型在代数几何”讨论上述成果所取得的卡西尼和他的合作者。与此同时，在这个国家的许多数学系举办了关于同一主题的研讨会。