Hodge Filtration, Singularities, and Complex Birational Geometry

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This project addresses problems of fundamental interest in pure mathematics, and especially in the field of algebraic geometry. Algebraic geometry is one of the oldest branches of mathematics, as people have attempted for a very long time to use algebra in order to understand problems in geometry, and at the same time one that has seen some of the most outstanding modern developments and connections with areas of pure and applied science. The main aim of the project is to connect parts of algebraic and complex geometry that have previously been quite disjoint; in particular, it uses new tools (called Hodge modules), relying heavily on topology and analysis as well, in order to classify geometric shapes and singularities. This approach is generating numerous projects for Ph.D. students and the project provides research training opportunities for graduate students.In more detail, the PI will continue applying the theory of mixed Hodge modules to concrete problems in complex and birational geometry. He will also continue developing the theory of Hodge ideals, in collaboration with M. Mustata. This has been completed for Q-divisors, providing an extension of the theory of multiplier ideas in this setting, but new ideas need to be brought into play in order to obtain a similar picture for ideal sheaves, or for local cohomology. One hopes that this will lead to further interesting applications. 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 will 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 will extend this study to families of singular varieties, especially those that appear in the theory of moduli of higher dimensional varieties according to Kollár and others, perhaps using those Hodge modules that extend variations of mixed Hodge structure. The PI will also continue working towards the classification 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.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.