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Hodge Theory and Birational Geometry

霍奇理论和双有理几何

基本信息

批准号：
1700819
负责人：
Mihnea Popa
金额：
$ 31万
依托单位：
Northwestern University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700819&HistoricalAwards=false
关键词：
Hodge Theory Birational Geometry

项目摘要

This project addresses problems of fundamental interest in the field of algebraic geometry. Algebraic geometry is one of the oldest fields in mathematics, as people have attempted for a very long time to use algebraic tools to understand problems in geometry, and at the same time it is a field that has seen some of the most outstanding modern developments and connections with areas of pure and applied science. This project will connect parts of algebraic and complex geometry that have previously been quite disjoint; in particular, it uses new tools (called Hodge modules) to classify geometric shapes and singularities. This approach will generate numerous projects for doctoral students. The PI will continue to be involved in outreach activities, like work towards creating a better environment for women in mathematics through his involvement in the Graduate Research Opportunities for Women (GROW) program at Northwestern. In more detail, the PI will continue to apply the theory of mixed Hodge modules to concrete problems in complex and birational geometry. He will pursue the development of the theory of Hodge ideals associated to divisors on smooth varieties, and its applications. This has been completed for reduced divisors, but significant new ideas will need to be brought into play in order to obtain a similar picture for Hodge ideals associated to Q-divisors or ideal sheaves. While Hodge ideals associated to reduced hypersurfaces are already a generalization of certain types of multiplier ideals, once this program is achieved, Hodge ideals will provide an enhancement of the theory of multiplier ideals in full generality. Consequently, one hopes for applications that reflect this. In recent work the PI gave applications regarding the singularities of theta divisors, and of hypersurfaces in projective space or toric varieties. In addition to further applications along these lines, the PI will use the proposed extensions to study Fujita-type problems, especially regarding the very ampleness of adjoint linear series, and also problems in local algebra. The PI has also been involved in applying the theory of Hodge modules to the study of families of smooth projective varieties, e.g. hyperbolicity questions in the sense of Viehweg. He will extend this study to families of singular varieties, especially those that appear in the theory of moduli of higher dimensional varieties, perhaps using those Hodge modules that extend variations of mixed Hodge structure. Finally, the PI will also continue working towards the classification of subvarieties with minimal cohomology class on principally polarized abelian varieties.

该项目解决代数几何领域的根本问题。代数几何是数学中最古老的领域之一，因为人们长期以来一直尝试使用代数工具来理解几何问题，同时它也是一个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联系的领域。该项目将连接以前相当不相交的代数和复杂几何部分；特别是，它使用新工具（称为 Hodge 模块）对几何形状和奇点进行分类。这种方法将为博士生带来大量项目。 PI 将继续参与外展活动，例如通过参与西北大学的女性研究生研究机会 (GROW) 项目，努力为女性数学领域创造更好的环境。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。他将致力于发展与平滑簇除数相关的霍奇理想理论及其应用。这已经针对约除数完成了，但是需要引入重要的新想法才能获得与 Q 因数或理想滑轮相关的霍奇理想的类似图像。虽然与简化超曲面相关的霍奇理想已经是某些类型的乘子理想的概括，但一旦实现该计划，霍奇理想将提供乘子理想理论的全面普遍性的增强。因此，人们希望应用程序能够反映这一点。在最近的工作中，PI 给出了关于 theta 约数的奇点以及射影空间或复曲面簇中的超曲面的应用。除了这些方面的进一步应用之外，PI 将使用所提出的扩展来研究 Fujita 型问题，特别是关于伴随线性级数的丰富性以及局部代数中的问题。 PI 还参与了将 Hodge 模理论应用于平滑射影簇族的研究，例如Viehweg 意义上的双曲问题。他将把这项研究扩展到奇异簇族，特别是那些出现在高维簇模理论中的簇，也许会使用那些扩展混合霍奇结构变体的霍奇模。最后，PI 还将继续致力于对主要极化的阿贝尔品种具有最小上同调类的亚品种进行分类。