Birational Geometry, Hodge Theory and Singularities

This project addresses problems of fundamental interest in pure mathematics, and especially in the field of algebraic geometry. This area of study is one of the oldest in mathematics, but also one of the most active currently. The past decade or so has seen some of the most outstanding modern developments and connections with areas of pure and applied science. The purpose of this project is to continue one such modern development, namely the application of methods based on the theory of so-called mixed Hodge modules to questions in higher dimensional complex geometry. These objects are the outcome of an intricate mix of algebra, analysis, and topology, and can be used to prove new results about geometric shapes and singularities. In particular, the PI develop new results about basic invariants of geometric objects (some of the key words here are the Kodaira dimension, or the local cohomological dimension). In the broader sense, the PI is involved with the mathematical community through his work on editorial boards, scientific boards and AMS committees, and through his lectures and expository notes prepared for US and international events. This project will provide research training opportunities for students. In more detail, the PI will continue studying questions in complex birational geometry and singularity theory, especially through the use of Hodge theory and D-modules. He will make further progress on the study of the Hodge filtration on the local cohomology of arbitrary subschemes of smooth varieties, and on the closely related theory of higher Du Bois and higher rational singularities. The general context requires significant new ideas compared to in the case of local complete intersections, which is by now rather well understood. This program will lead to new applications, similar to what the theory of multiplier ideals, and more generally Hodge ideals, produced in the case of hypersurfaces. In a different direction, the PI has recently proposed a conjecture on the superadditivity of the Kodaira dimension for morphisms between smooth complex projective varieties, and more generally an additivity conjecture for smooth projective morphisms between quasi-projective varieties; they complement Iitaka's well-known subadditivity conjecture, a problem of central importance in birational geometry. Some results have already been obtained, and the PI will make further progress on these conjectures, for instance by showing that additivity holds when the general fiber of the morphism admits a good minimal model. This is closely related to other interesting projects, for instance studying a natural generalization of Viehweg's hyperbolicity conjecture in the same setting. The PI proposes to attack further problems in algebraic dynamics, and in the analytic study of the V-filtration. 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.