Hodge Filtration, Singularities, and Complex Birational Geometry

这个项目解决了在纯数学，特别是在代数几何领域的基本兴趣问题。代数几何是数学最古老的分支之一，因为人们在很长一段时间里都试图用代数来理解几何问题，同时它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代数几何和复杂几何的部分连接起来，这些部分以前是完全脱节的；特别是，它使用了新的工具（称为Hodge模块），主要依赖于拓扑和分析，以便对几何形状和奇点进行分类。这种方法为博士生产生了大量的项目，并为研究生提供了研究训练的机会。更详细地说，PI将继续将混合Hodge模块理论应用于复杂几何和双几何的具体问题。他还将与穆斯塔塔先生合作，继续发展霍奇理想理论。对于q -除数，这已经完成了，在这种情况下提供了乘子思想理论的扩展，但是为了获得理想束或局部上同的类似图像，需要引入新的思想。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已经获得了关于因子奇点、射影空间中的超曲面或最小指数的应用。除了沿着这些路线进一步的结果，他们将使用提出的扩展来研究，例如，线性级数的有效界，或伯恩斯坦-佐藤多项式的根。PI还参与了将Hodge模块理论应用于研究族的光滑投影变体的变化，例如Brody双曲或参数空间的viehweg型问题。他将把这项研究扩展到奇异品种的族，特别是那些根据Kollár和其他人在高维品种的模理论中出现的族，也许使用那些扩展混合Hodge结构变化的Hodge模块。PI还将继续研究在主极化阿贝尔变体上具有最小上同调类的子变体的分类，以及它与一般消失子格式和与奇异性的联系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.