Hodge Theory and Birational Geometry

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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.