Higher Multiplier Ideals and Other Applications of Hodge Theory in Algebraic Geometry

This is a project in pure mathematics, more precisely in the field of algebraic geometry. Algebraic geometry studies geometric objects that are defined by polynomial equations, using methods from algebra, complex analysis, differential geometry, topology, and partial differential equations. Algebraic geometry also provides useful examples for certain parts of theoretical physics such as string theory. Most of the research proposed by the PI has to do with a subfield of algebraic geometry called Hodge theory. Hodge theory makes it possible to apply results from the theory of partial differential equations to problems in algebraic geometry, in a language that is closer to algebra than to analysis. Some of the work proposed by the PI is about developing Hodge theory further (and to make it more accessible to graduate students); the rest is about using Hodge theory to solve several very specific problems in algebraic geometry. The PI will also work on a comprehensive book on this subject which will be of enormous use to the research community. More specifically, this project has several different research objectives, all related to Hodge theory: (1) To develop a theory of higher multiplier ideals for effective divisors, in joint work with Ruijie Yang, by using the nearby cycles functor for mixed Hodge modules. (2) To apply this theory to theta divisors on principally polarized abelian varieties, in particular to a conjecture by Casalaina-Martin about the multiplicities of their singular points. (3) To continue to study the local and global structure of the locus of self-dual classes for integral variations of Hodge structure; self-dual classes are a generalization of Hodge classes that are of interest in theoretical physics. (4) To further investigate the behavior of Kodaira dimension under smooth morphisms, especially several related conjectures by Campana-Peternell and by Popa. (5) To continue the study of degenerating complex variations of Hodge structure in higher dimensions, in particular the precise local behavior of the Hodge metric, and its relation with the representation theory of semisimple Lie algebras. (6) To continue the "Mixed Hodge Module Project" (joint with Claude Sabbah), whose aim is to write an accessible treatment of the theory of mixed Hodge modules with complex coefficients. 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.