Derived Equivalences, Generic Vanishing, and the Structure of Cohomology

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

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 field, create interaction with people of different 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.