Hodge Filtration, Singularities, and Complex Birational Geometry

Overview. The PI proposes to continue applying the theory of mixed Hodge modules to concrete problems in complex and birational geometry. Continuing his long-term project with M. Mustata, he intends to pursue the development of the theory of Hodge ideals, as well as of the Hodge fi�ltration on the local cohomology, both associated to arbitrary subschemes of smooth varieties. This requires signi�ficant new ideas compared to the study of Hodge ideals associated to Q-divisors, which has recently been completed. Once this program is achieved, the Hodge fi�ltration on local cohomology will provide an enhancement of the theory of multiplier ideals in its full generality. Consequently, he intends to provide applications that reflect this. 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 are planning to 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 would like to extend this study to families of singular varieties, especially those that appear in the theory of moduli of higher dimensional varieties according to Kollar and others, perhaps using those Hodge modules that extend variations of mixed Hodge structure. The PI would also like to continue working towards the classi�fication 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. Intellectual merit. Some of the problems the PI proposes to attack, like the study of roots of Bernstein-Sato polynomials, effective bounds in birational geometry, hyperbolicity of parameter spaces of varieties, subvarieties of minimal class in abelian varieties, or singularities of theta divisors, are among the most prominent problems in their respective areas and will have a high impact as proved statements. Others, especially the development of the theory of Hodge ideals or of the Hodge �filtration on local cohomology are part of newly emerging directions of research, where a deeper understanding is guaranteed to lead to more applications. All parts of the project will have a broad range of applications, further our knowledge in the �field, lead to interaction with experts in other areas, and produce problems suitable for students. The work of my Ph.D. students already contains results directly related to the problems and techniques proposed here. Broader impacts. The PI will continue to contribute towards improving the environment for women in mathematics through his mentorship at both the post-graduate and graduate level (and hopefully undergraduate as well), as a follow-up to his participation in the GROW program at Northwestern. In the international mathematical community, he has been and will continue to be involved in organizing conferences and workshops, editing expository volumes, and continuing work on editorial (like the Journal of Algebraic Geometry, and Algebraic Geometry until this year) and scientifi�c (Superior Normal School, Bucharest) boards. The PI will continue to deliver lectures at summer schools and conferences in the U