Algebraic geometry is the study of solutions of polynomial equations. This is a central area of mathematics with a vast range of applications, a small sampling of which includes physics, cryptography, and computational complexity (the design of faster, more efficient computational algorithms). A powerful tool in the subject is Hodge theory, which associates a linear algebraic structure to a set of solutions of a system of polynomial equations. The advantage of this association is that the Hodge structure is often more amenable to computation and analysis than the original set of solutions, while at the same time retaining enough information to yield deep insights into the set of solutions (and the questions that one would like to answer about them). The principal objective of this research project is to develop Hodge theory so that it can be applied to a greater range of problems in algebraic geometry.
The project will address a collection of problems in complex geometry and Lie theory that are motivated by Hodge theory. In practice this means that many of the questions concern (generalized) flag manifolds and flag domains, a canonical system of geometric partial differential equations on these spaces (whose integrals are called horizontal submanifolds), and locally homogeneous spaces. In Hodge theory, flag domains arise as period domains, the classifying spaces for polarized Hodge structures; flag manifolds as compact duals of period domains; the canonical system of geometric PDE as the infinitesimal period relation, Griffiths' system of differential equations constraining variations of Hodge structure. The work will include: (1) a generalization of the Satake-Baily-Borel compactification and Borel's extension theorem for locally Hermitian symmetric spaces to a class of locally homogeneous spaces of Hodge theoretic interest; (2) investigation of a conjectural mixed Hodge structure on the characteristic cohomology of the infinitesimal period relation; and (3) initiation of a program to study the extent to which a variation of Hodge structure is determined by its characteristic varieties.
