Hodge Theory and Birational Geometry

这个项目解决了代数几何领域的基本问题。代数几何是数学中最古老的领域之一，因为人们已经尝试了很长时间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和与纯科学和应用科学领域的联系的领域。这个项目将连接代数几何和复杂几何的部分，这些部分以前是完全脱节的；特别是，它使用新的工具（称为Hodge模块）来对几何形状和奇点进行分类。这种方法将为博士生产生大量的项目。PI将继续参与外展活动，比如通过参与西北大学的女性研究生研究机会（GROW）项目，为女性创造更好的数学环境。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中的具体问题。他将继续发展霍奇理想理论，并将其应用于光滑变量的除数及其应用。这已经完成了约化除数，但为了获得与q除数或理想束相关的霍奇理想的类似图像，还需要引入重要的新思想。虽然与简化超曲面相关的Hodge理想已经是某些类型乘法器理想的概括，但一旦这个程序实现，Hodge理想将提供乘法器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给出了关于因子的奇异性的应用，以及射影空间或环面变体中的超曲面。除了沿着这些路线的进一步应用之外，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非常丰度的问题，以及局部代数问题。PI还参与了将Hodge模块理论应用于光滑投影变量族的研究，例如Viehweg意义上的双曲问题。他将把这项研究扩展到奇异变种的家族，特别是那些出现在高维变种的模理论中的家族，也许使用那些扩展混合Hodge结构变化的Hodge模块。最后，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。

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.