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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

(p.45) Chapter II Period Domains and Mumford-Tate Domains
Mumford-Tate Groups and Domains

Mark Green

Phillip Griffiths

Matt Kerr

Princeton University Press

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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