This mathematical research project concerns algebraic geometry and related topics. Some of the problems under study in the project involve the extension of tools of algebraic geometry using algebraic structures called Hodge modules. Others involve checking invariance of various topological quantities under derived equivalences. All parts of the project will have a broad range of applications, further knowledge in the field, initiate interaction among people of different mathematical backgrounds, and produce problems suitable for research involvement of Ph.D. students.
This project studies cohomological, numerical, and singularity invariants of projective manifolds by looking at their derived categories of coherent sheaves, and by applying techniques from generic vanishing and mixed Hodge module theory. The PI would like to develop a vanishing, injectivity and extension package to be added to Saito's Kodaira-type vanishing theorem for Hodge modules, and to use this in order to attack problems on the variation of families of varieties of varieties of general type or Kodaira dimension zero. He is also interested in studying generalizations of multiplier ideals coming from the Hodge filtration on localized D-modules, and to use this for attacking a conjecture on singularities of theta divisors. Another aim is to study singularities in the minimal model program by establishing a connection between Saito's theory and the filtered de Rham complex of singular varieties. In the direction of derived categories, the main topic under study is comparison of the cohomological invariants and the geometry of varieties with equivalent bounded derived categories of coherent sheaves, a topic of interest both in mirror symmetry and in birational geometry. Continuing work on the behavior of the Picard variety and of certain Hodge numbers under derived equivalence, the PI plans to address problems like the invariance of the canonical cohomology and the invariance of cohomological support loci coming from generic vanishing theory. He is also interested in studying similar questions in the singular setting, extending some previous work in the case of varieties with quotient singularities.
