Arithmetic Study Group: Mumford-Tate groups and the classification of Hodge structures
12 January 2010 15:15 in CM219
Since their introduction in the mid-20th Century, Hodge structures have
been a fundamental tool in transcendental algebraic geometry, for example
in the study of algebraic cycles and moduli of complex algebraic varieties. Mumford-Tate groups are the symmetry groups of Hodge theory,
and their orbits (Mumford-Tate domains) are the moduli spaces for Hodge
structures with given symmetries.
The 'classical' case of Hodge structures of weight 1 (and those they generate by linear-algebraic constructions) has been thoroughly studied.
In this case, the MT-domains are Hermitian symmetric spaces whose arithmetic quotients yield algebraic (Shimura) varieties. The many beautiful results facilitated by MT groups in this setting include Deligne's theorem on absolute Hodge cycles and the resolution (by many authors) of the full Hodge conjecture for various classes of abelian varieties.
Following on a review of this history, I will describe recent joint work
with P. Griffiths and M. Green on the "nonclassical" higher weight case.
The corresponding theory is in its early stages and is of an entirely different character: Shimura varieties are replaced by global integral
manifolds of an exterior differential system, and nonclassical (exceptional) Lie groups turn out to occur as MT groups. In addition to
the general context mentioned above, part of the motivation for our project was to better understand the very interesting special features of period domains associated to Calabi-Yau 3-folds, and I will explain a classification result for the MT subdomains in an important special case.