A central problem originating on the homological side of mirror symmetry and birational geometry is to compare the numerical invariants, cohomology, and geometry of varieties with equivalent bounded derived categories. Continuing work with Schnell on the behavior of the Picard variety under derived equivalence and the invariance of certain Hodge numbers, the PI plans to address problems like the invariance of the canonical cohomology, the invariance of cohomological support loci, and the invariance of all Hodge numbers for Fourier-Mukai partners, or the study of the dimension of derived categories of coherent sheaves. In a different direction, many results on the cohomology of compact Kahler manifolds lie at the intersection of the areas of Generic Vanishing theory, Fourier-Mukai theory, and homological and commutative algebra. The PI will continue work in this area, initiated with Lazarsfeld, on the structure of the canonical cohomology as a module over the exterior algebra. He would like to derive consequences on either the existence of indecomposable bundles of low rank on projective space, or on improved lower bounds on the holomorphic Euler characteristic of compact Kahler manifolds without irregular fibrations, after applying the BGG correspondence to such modules. He will attempt to apply methods of this type to approach a conjecture of Carrell and Hacon-Kovacs on holomorphic one-forms on varieties of general type. He will also continue work towards the Beauville-Debarre-Ran Schottky-type conjecture predicting which principally polarized abelian varieties contain subvarieties representing minimal cohomology classes, and towards the Beauville conjecture on filtrations on the rational Chow groups of abelian varieties induced by the Fourier-Mukai transform.
Some of the problems the PI proposes to attack, like the invariance of cohomology groups or Hodge numbers under derived equivalences, restrictions on the total cohomology of a variety, or the study of minimal classes on abelian varieties, are among the most prominent and established problems in their respective areas and will have a high impact as proved statements. Others, like for instance the dimension of derived categories or the exterior structure on cohomology modules, are part of newly emerging theories where a better understanding will surely lead to more applications. All parts of the project will have a broad range of applications, further our knowledge in the fi eld, create interaction with people of diff^erent mathematical backgrounds, and produce problems suitable for Ph.D. students. Outside of Mathematics, the PI will continue to be involved in non-departmental activities, like his membership on the WISEST Committee devoted to creating a better environment for women in science and engineering, and on the UIC Senate. In the international mathematical community, he will be involved in organizing conferences and workshops, and editing volumes with the goal of disseminating knowledge. Funds will help the PI continue to deliver lectures at summer schools and conferences in the US and abroad; in the recent past this has led to working with students outside of the PI's base institution and getting part of their research started. In the UIC department they will be used for assisting the research and travel of graduate students, supporting seminars, and developing and improving the graduate curriculum.
The research: In work supported by this grant, the PI has extended the main results of Generic Vanishing theory to the setting of mixed Hodge modules, and used this to prove generic Nakano-type vanishing and to confirm a conjectureon the non-existence of nowhere vanishing holomorphic one-forms on varieties of general type. He has also made progress towards proving the invariance of cohomological support loci for derived equivalent irregular varieties, and of orbifold Hodge numbers for derived equivalent smooth Deligne-Mumford stacks.Part of this work has been carried out together with L. Lombardi (Univ. of Bonn) and C. Schnell (Stony Brook University). For the Generic Vanishing project, the fundamental point has been to extend the scope of the theory to filtered D-modules arising from Hodge theory, more precisely those underlying mixed Hodge modules. This has allowed us to bring into play M. Saito's powerful theory, in combination with a Fourier-Mukai transform for D-modules on abelian varieties introduced by Laumon and Rothstein. We were able to extend all of the main results of Generic Vanishing theory due to Ein-Lazarsfeld, Hacon, Pareschi and the PI, Arapura and Simpson to this setting, and this in turn allowed us to obtain new applications to the study of fundamental objects like bundles of holomorphic forms, rank one local systems, or holomorphic one-forms. For the derived equivalences project, the driving problem has been that of the invariance of Hodge numbers for Fourier-Mukai partners. In the case of irregular varieties, the PI discovered that the most powerfulapproach is to extend the scope of the problem to proving the invariance of more refined objects called cohomological support loci. A proper phrasing of this conjectural invariance addressed the Hodge numbers problem as well. We were able to prove this conjecture in dimension up to three, and make further progress in arbitrary dimension. Intellectual merit and broader impacts: Problems that the PI has addressed in the various projects supported by this NSF grant, like extensions of Generic Vanishing theory, the behavior of zeros of holomorphic one-forms, or the invariance of numerical and geometric invariantsunder derived equivalence, are among the most prominent and established problems in their respective areas,and have already had visible impact. For instance, the extension of generic vanishing theorems to mixed Hodge modules has already been used by Budur, Wang, Schnell and the PI to approach other basic problems related to local systems or birational geometry. All parts of the project have created interaction with people of different mathematical backgrounds, and produced problems suitable for Ph.D. students; the PI has a few new students working in these areas. Outside of mathematics, the PI has been involved in non-departmental activities, like events devoted to raising the profile of women in mathematics and science. In the international mathematical community, he has been involved in organizing conferences and workshops, has edited volumes with the goal of disseminating knowledge, and has delivered lectures at summer schools and conferences. Together with a few others, he has started the organization of the largest Algebraic Geometry conference to be held over the last decade. In the UIC department he has assisted the research of graduate students, supported seminars, and has been involved in developing and improving the graduate curriculum.
