Hodge Filtration, Singularities, and Complex Birational Geometry

这个项目解决了纯数学中基本感兴趣的问题，特别是在代数几何领域。代数几何是数学中最古老的分支之一，因为很长一段时间以来，人们一直试图使用代数来理解几何中的问题，同时也看到了一些最杰出的现代发展以及与理论科学和应用科学领域的联系。该项目的主要目标是将代数和复杂几何中以前非常不相交的部分联系起来；特别是，它使用了新的工具(称为Hodge模块)，严重依赖于拓扑和分析，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，该项目为研究生提供了研究培训的机会。更详细地说，PI将继续将混合霍奇模块的理论应用于复杂和双曲面几何中的具体问题。他还将与M·穆斯塔塔合作，继续发展霍奇理想理论。这对于q-因子来说已经完成了，提供了乘子思想理论在这个背景下的扩展，但是需要引入新的思想，以获得理想层或局部上同调的类似的图景。人们希望这将导致更多有趣的应用。在他们的工作中，PI和Mustata已经得到了关于theta因子的奇异性、射影空间中的超曲面或最小指数的应用。除了沿着这些路线的进一步结果，他们将使用所提出的扩展，以研究，例如，线性级数的有效界，或伯恩斯坦-佐藤多项式的根。PI还将Hodge模的理论应用到变种的族光滑射影变种的研究中，例如关于参数空间的Brody双曲性或Viehweg型问题。他将把这项研究扩展到奇异变种的家族，特别是那些出现在Kollár和其他人的高维变种的模理论中的变种，也许使用那些扩展混合Hodge结构的变种的Hodge模。PI还将继续致力于对主要极化阿贝尔品种上具有最小上同源类别的亚品种进行分类，并将其与通用消失子方案和theta因子的奇异性联系起来。该奖项反映了NSF的法定使命，并已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。

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.