Hodge Filtration, Singularities, and Complex Birational Geometry

霍奇过滤、奇点和复杂双有理几何

• 批准号: 2000610
• 负责人: Mihnea Popa
• 金额: $ 33.6万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000610&HistoricalAwards=false
• 关键词: Hodge; Filtration; Singularities; Complex; Birational

This project addresses problems of fundamental interest in pure mathematics, and especially in the field of algebraic geometry. Algebraic geometry is one of the oldest branches of mathematics, as people have attempted for a very long time to use algebra in order to understand problems in geometry, and at the same time one that has seen some of the most outstanding modern developments and connections with areas of pure and applied science. The main aim of the project is to connect parts of algebraic and complex geometry that have previously been quite disjoint; in particular, it uses new tools (called Hodge modules), relying heavily on topology and analysis as well, in order to classify geometric shapes and singularities. This approach is generating numerous projects for Ph.D. students and the project provides research training opportunities for graduate students.In more detail, the PI will continue applying the theory of mixed Hodge modules to concrete problems in complex and birational geometry. He will also continue developing the theory of Hodge ideals, in collaboration with M. Mustata. This has been completed for Q-divisors, providing an extension of the theory of multiplier ideas in this setting, but new ideas need to be brought into play in order to obtain a similar picture for ideal sheaves, or for local cohomology. One hopes that this will lead to further interesting applications. In their work the PI and Mustata have already obtained applications regarding the singularities of theta divisors, hypersurfaces in projective space, or minimal exponents. In addition to further consequences along these lines, they will use the proposed extensions in order to study, for instance, effective bounds for linear series, or roots of the Bernstein-Sato polynomial. The PI has also been involved in applying the theory of Hodge modules towards the study of the variation of families smooth projective varieties of varieties, e.g. Brody hyperbolicity or Viehweg-type questions for parameter spaces. He will extend this study to families of singular varieties, especially those that appear in the theory of moduli of higher dimensional varieties according to Kollár and others, perhaps using those Hodge modules that extend variations of mixed Hodge structure. The PI will also continue working towards the classification of subvarieties with minimal cohomology class on principally polarized abelian varieties, and its link with generic vanishing subschemes and with the singularities of theta divisors.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目解决了纯数学兴趣的基本兴趣问题，尤其是在代数几何学领域。代数几何形状是数学最古老的分支之一，因为人们试图使用代数来了解几何学的问题，同时又看到了一些最杰出的现代发展和与纯科学领域的联系。该项目的主要目的是连接代数和复杂几何形状的一部分，这些几何形状以前是不相交的。特别是，它使用新工具（称为Hodge模块），在很大程度上依赖于拓扑和分析，以对几何形状和奇异性进行分类。这种方法正在为博士生成许多项目。学生和该项目为研究生提供了研究培训机会。更详细的是，PI将继续将混合Hodge模块的理论应用于复杂和哲学的具体问题。他还将与M. Mustata合作发展Hodge思想的理论。这已经为Q划分人完成，在此环境中提供了乘数思想理论的扩展，但是需要发挥新的想法，以便为理想的滑轮或本地共同体学获得类似的图片。希望这将导致进一步的有趣应用程序。在他们的工作中，Pi和Mustata已经获得了有关Theta除数，投射空间中的超曲面或最小指数的唯一申请。除了沿着这些线路的进一步后果外，它们还将使用所提出的扩展来研究线性序列的有效界限，或者是伯恩斯坦 - 西托·多项式的根。 PI还参与了将Hodge模块理论应用于研究家族的平滑射击变化变化的变化，例如Brody双曲度或参数空间的ViewWeg型问题。他将将这项研究扩展到奇异品种的家族，尤其是那些根据Kollár和其他人的模型理论出现的，也许使用了那些扩展混合霍奇结构变化的Hodge模块。 PI还将继续致力于对主要两极分化的Abelian品种的最低限度同胞学班级的分类，并与通用消失的亚种和Theta Divides的奇异性联系起来。该奖项反映了NSF的法规任务，并通过对基金会的知识优点和广泛的criitia进行评估，以评估为您提供了珍贵的支持。