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Mumford-Tate Groups and DomainsTheir Geometry and Arithmetic (AM-183)$
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Mark Green, Phillip A. Griffiths, and Matt Kerr

Print publication date: 2012

Print ISBN-13: 9780691154244

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691154244.001.0001

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Period Domains and Mumford-Tate Domains

Period Domains and Mumford-Tate Domains

Chapter:
(p.45) Chapter II Period Domains and Mumford-Tate Domains
Source:
Mumford-Tate Groups and Domains
Author(s):

Mark Green

Phillip Griffiths

Matt Kerr

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691154244.003.0003

This chapter provides an introduction to the basic definitions of period domains and their compact duals as well as the canonical exterior differential system on them. The period domain D is comprised of a set of polarized Hodge structures. The natural symmetry group acting on D is the group G(ℝ) of real points of the ℚ-algebraic group G = Aut(V,Q). Elementary linear algebra shows that G(ℝ) operates transitively on D. The chapter also discusses Mumford-Tate domains and their compact duals as well as the Noether-Lefschetz locus in period domains. The basic properties of Mumford-Tate domains are established in several places.

Keywords:   period domain, compact dual, polarized Hodge structure, natural symmetry group, Mumford-Tate domain, Noether-Lefschetz locus

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