Algebraic Geometry Seminar: "Identification of Hodge domains"

Abstract: Period domains parameterize polarized Hodge structures (PHS).  Hodge domains are distinguished (complex, homogeneous) submanifolds that parameterize PHS with nongeneric Hodge tensors.  One is interested in identifying these subdomains for a number of reasons:  First, when the PHS is realized by the cohomology of an algebraic variety, the variety should admit nongeneric arithmetic properties (such as extra Hodge classes, or automorphisms, et cetera).  Second, the horizontal Hodge domains D are Hermitian symmetric.  Consequently, both D and any arithmetic quotient X admits a variety of compactifications.  Given a moduli space geometrically realizing as a Hodge domain (with a Torelli theorem), we may then obtain an induced compactification of the moduli space.

I will review the theoretical framework (Hodge representations) developed by Green, Griffiths and Kerr to identify Hodge domains, illustrate its application in some simple cases and discuss some open problems.

Host: Matt Kerr