Abelian varieties are at the same time algebraic varieties (spaces described by polynomial equations) and abelian groups (sets with an operation that obeys the same laws as addition on the set of whole numbers), and what makes them interesting to a mathematician is the interplay between those two structures. They have been intensively studied since the 19th century, especially in the field of algebraic geometry, but many important questions remain unsolved. This project will take a new approach to some of those questions, by investigating the properties of D-modules on abelian varieties. D-modules are basically systems of partial differential equations, in an abstract form that is useful for studying geometric questions. For example, whenever one has a mapping from another algebraic variety to an abelian variety, one gets a certain number of D-modules on the abelian variety that contain information about the original mapping. The main new idea is that interesting properties of these (and other) D-modules on abelian varieties can be revealed with the help of Fourier analysis, in the same way that interesting properties of signals (such as sound waves) can be revealed by looking at their frequency spectrum. The hope is that a good understanding of the "spectrum" of D-modules on abelian varieties will lead to new results about their geometry.
In more technical language, the research objective of the project is to give a complete characterization of Fourier-Mukai transforms of holonomic D-modules on complex abelian varieties. These transforms are complexes of coherent sheaves on the moduli space of line bundles with connection (which is a hyperkaehler manifold), and in many ways, they look remarkably similar to perverse sheaves. This suggests that Fourier-Mukai transforms of holonomic D-modules should make up an as-yet conjectural category of "hyperkaehler perverse sheaves"; the project will make this idea precise and test some of its implications. This question is also of practical interest, because it provides a new tool for solving problems about abelian varieties and irregular varieties, similar to the generic vanishing theorem of Green and Lazarsfeld. A second objective is to make the theory of mixed Hodge modules more widely known. Building on a recent Clay Mathematics Institute workshop, the PI and others plan to write a book about mixed Hodge modules and their applications that will try to make this powerful theory accessible to non-experts.
