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Hodge Filtration, Singularities, and Complex Birational Geometry

霍奇过滤、奇点和复杂双有理几何

基本信息

批准号：
2040378
负责人：
Mihnea Popa
金额：
$ 33.6万
依托单位：
Harvard 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 2025-04-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2040378&HistoricalAwards=false
关键词：
Hodge Filtration Singularities Complex Birational

项目摘要

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.

该项目解决纯数学中的根本问题，尤其是代数几何领域的问题。代数几何是数学最古老的分支之一，因为人们已经尝试了很长一段时间来使用代数来理解几何问题，同时也看到了一些最杰出的现代发展和与纯科学和应用科学领域的联系。该项目的主要目的是将代数和复杂几何中以前不相交的部分连接起来;特别是，它使用了新的工具（称为霍奇模块），严重依赖拓扑和分析，以便对几何形状和奇点进行分类。这种方法产生了许多博士项目。该项目为研究生提供了研究培训机会。更详细地说，PI将继续将混合霍奇模理论应用于复杂和双有理几何的具体问题。他还将继续发展霍奇理想的理论，与M。穆斯塔塔这已经完成了Q-因子，提供了一个扩展的理论乘数的想法在这种情况下，但新的想法需要发挥作用，以获得类似的图片理想层，或局部上同调。人们希望这将导致进一步有趣的应用。在他们的工作中，PI和Mustata已经获得了关于θ因子的奇异性、射影空间中的超曲面或最小指数的应用。除了进一步的后果沿着这些线，他们将使用建议的扩展，以研究，例如，有效的界限线性系列，或根的伯恩斯坦-佐藤多项式。PI还参与了应用霍奇模理论对研究家庭的变化顺利投影品种的品种，例如布罗迪双曲或Viehweg型问题的参数空间。他将这项研究扩展到家庭的奇异品种，特别是那些出现在理论的模的高维品种根据Kollár和其他人，也许使用这些霍奇模块，延长变化的混合霍奇结构。PI还将继续致力于对主要极化阿贝尔品种的最小上同调类的子品种的分类，以及其与通用消失子方案和theta因子奇异性的联系。该奖项反映了NSF的法定使命，并被认为是值得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的支持。