Hodge Theory, Moduli, and Representation Theory

Focused Research Group, June 1 2014 -- July 31 2017

Hodge theory is a central area of algebraic geometry with roots in the the classical (19th century) theory of special functions and period integrals. From a modern point of view, the goal of Hodge theory is to relate topological invariants of algebraic varieties to arithmetic and analytic invariants. The central notion is that of a Hodge structure on the cohomology groups of an algebraic variety. While the cohomology groups are purely topological, depending only on the shape of variety, the Hodge structure is a much more sensitive invariant. Consequently, the Hodge structure carries a great deal of important algebro-geometric and number-theoretical information.

The most famous unsolved problem in algebraic geometry is the Hodge conjecture, a question about the relationship between the Hodge structure of the cohomology groups of a variety and the existence of certain subvarieties. This focus on the relationship between topological objects and finer analytic invariants is typical of Hodge theory as a whole, and it is the main motivation for the research supported by this FRG. We expect that this research will consequently impact several areas of mathematics including number theory, algebraic geometry and representation theory.
The FRG members intend to focus on four related topics: (1) Mumford-Tate (MT) domains, (2) moduli spaces, (3) algebraic cycles and the Hodge conjecture, and (4) mixed Hodge modules.

(1) MT domains are classifying spaces of Hodge structures, and, roughly speaking, the boundary components of Mumford-Tate domains parametrize degenerations of Hodge structures. The PIs intend to advance number theory, representation theory and algebraic geometry by studying Mumford-Tate domains and their boundary components. For example, the PIs plan to extend work of Carayol, which seeks to associate Galois representations to automorphic representations whose archimedian component is a degenerate limit of discrete series.

(2) The second topic concerns the realization of moduli spaces of geometric objects as quotients by discrete groups. An example of such a realization is the moduli space of non-hyperelliptic genus 3 curves, which can be realized as a ball quotient, where the 6 dimensional ball in question sits in the MT domain of K3 surfaces. However, there are not many examples of this type known. The PIs intend to look for more.

(3) The third topic involves the approach to the Hodge conjecture via normal functions and their singularities due to Green and Griffiths. The PIs will develop this approach in several directions. For example, they will study the archimedean height function associated to a normal function, and they intend to study the non-reductive MT groups associated to normal functions.

(4) Finally, the PIs will develop a flexible theory of complex variations of mixed Hodge modules and apply it to questions arising in representation theory. In particular, they would like to understand the structure of conformal blocks viewed as complex mixed Hodge modules on the moduli spaces of stable curves.
We gratefully acknowledge funding from the National Science Foundation under the aegis of DMS-1361159, DMS-1361120, DMS-1361147 and DMS-1361143.