Classification of Mumford-Tate Subdomains

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.