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，特别是代数几何领域的基本问题。代数几何是数学最古老的分支之一，因为人们长期以来一直尝试使用代数来理解几何问题，同时代数几何也见证了一些最杰出的现代发展以及与纯科学和应用科学领域的联系。该项目的主要目的是将以前相当不相交的代数和复杂几何部分连接起来；特别是，它使用新工具（称为 Hodge 模块），并严重依赖拓扑和分析，以便对几何形状和奇点进行分类。这种方法正在为博士生创造大量项目。更详细地说，PI 将继续将混合 Hodge 模理论应用于复杂和双有理几何的具体问题。他还将与 M. Mustata 合作，继续发展霍奇理想理论。这已经针对 Q-除数完成，在这种情况下提供了乘法器思想理论的扩展，但是需要运用新的思想才能获得理想滑轮或局部上同调的类似图像。人们希望这将带来更多有趣的应用。在他们的工作中，PI 和 Mustata 已经获得了有关 θ 约数奇点、射影空间中的超曲面或最小指数的应用。除了这些方面的进一步后果之外，他们还将使用所提出的扩展来研究线性级数的有效界限或伯恩斯坦-佐藤多项式的根等。 PI 还参与应用 Hodge 模块理论来研究家族的变异以及品种的平滑投影变种，例如参数空间的布罗迪双曲性或 Viehweg 型问题。他将把这项研究扩展到奇异簇族，特别是那些出现在 Kollár 等人的高维簇模理论中的簇，也许会使用那些扩展混合 Hodge 结构变体的 Hodge 模。 PI 还将继续致力于对主要极化阿贝尔品种的具有最小上同调类的亚品种进行分类，及其与通用消失子方案和 theta 除数奇异性的联系。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。