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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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Classification of Mumford-Tate Subdomains

Classification of Mumford-Tate Subdomains

(p.240) Chapter VII Classification of Mumford-Tate Subdomains
Mumford-Tate Groups and Domains

Mark Green

Phillip Griffiths

Matt Kerr

Princeton University Press

This chapter develops an algorithm for determining all Mumford-Tate subdomains of a given period domain. The result is applied to the classification of all complex multiplication Hodge structures (CM Hodge structures) of rank 4 and when the weight n = 1 and n = 3, to an analysis of their Hodge tensors and endomorphism algebras, and the number of components of the Noether-Lefschetz locus. The result is that one has a complex but very rich arithmetic story. Of particular note is the intricate structure of the components of the Noether-Lefschetz loci in D and in its compact dual, and the two interesting cases where the Hodge tensors are generated in degrees 2 and 4. One application is that a particular class of period maps appearing in mirror symmetry never has image in a proper subdomain of D.

Keywords:   period map, Mumford-Tate subdomain, period domain, complex multiplication Hodge structure, Hodge tensor, endomorphism algebra, Noether-Lefschetz locus, Hodge structure

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