Hodge theory and birational geometry

Overview. The PI proposes to continue applying the theory of mixed Hodge modules to concrete problems in complex and birational geometry. He intends to pursue the development of the theory of Hodge ideals associated to divisors on smooth varieties, and its applications, in collaboration with M. Mustata. This has been completed for reduced divisors, but significant new ideas need to be brought into play in order to obtain a similar picture for Hodge ideals associated to Q-divisors, or to ideal sheaves. While those 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 its full generality. Consequently, one hopes for applications that reflect this. In their work the PI and Mustata 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, they are planning to use the proposed extensions in order to study for instance Fujita-type problems, especially regarding the very ampleness of adjoint linear series, and also problems in local algebra. The PI has also been involved, together with C. Schnell, in applying the theory of Hodge modules towards the study of the variation of families smooth projective varieties of varieties, e.g. hyperbolicity questions in the sense of Viehweg. 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 classification of subvarieties with minimal cohomology class on principally polarized abelian varieties, and its link with generic vanishing subschemes, classified in dimension up to five in recent work with Casalaina-Martin and Schreieder. Intellectual merit. Some of the problems the PI proposes to attack, like the study of effective bounds for adjoint linear series, control on the singularities of theta divisors, the variation of families of varieties, or the classification of subvarieties of minimal class in abelian varieties, are among the most prominent and established problems in their respective areas and will have a high impact as proved statements. Others, especially the development of the theory of Hodge ideals, 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, create interaction with people of different mathematical backgrounds, 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 participate in outreach activities, like work towards creating a better environment for women in mathematics through his involvement 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 and scientific boards. For instance, last year he co-organized the AMS Summer Institute in Algebraic Geometry at University of Utah, and over this and the following year he will be involved in co-editing the