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This book deals with Mumford-Tate groups, the fundamental symmetry groups in Hodge theory. Much, if not most, of the use of Mumford-Tate groups has been in the study of polarized Hodge structures of level one and those constructed from this case. In this book, Mumford-Tate groups M will be reductive algebraic groups over ℚ such that the derived or adjoint subgroup of the associated real Lie group M_ℝ contains a compact maximal torus. In order to keep the statements of the results as simple as possible, the book emphasizes the case when M_ℝ itself is semi-simple. The discussion covers period domains and Mumford-Tate domains, the Mumford-Tate group of a variation of Hodge structure, Hodge representations and Hodge domains, Hodge structures with complex multiplication, arithmetic aspects of Mumford-Tate domains, classification of Mumford-Tate subdomains, and arithmetic of period maps of geometric origin.

This chapter deals with Hodge representations and Hodge domains. For general polarized Hodge structures, it considers which semi-simple ℚ-algebraic groups M can be Mumford-Tate groups of polarized Hodge structures, the different realizations of M as a Mumford-Tate group, and the relationship among the corresponding Mumford-Tate domains. The chapter uses standard material from the structure theory of semisimple Lie algebras and their representation theory. The discussion covers the adjoint representation and characterization of which weights give faithful Hodge representations, the classical groups and the exceptional groups, and Mumford-Tate domains as particular homogeneous complex manifolds. The examples concerning the classical groups illustrate both the linear algebra and Vogan diagram methods.

This chapter describes the arithmetic aspects of Mumford-Tate domains and Noether-Lefschetz loci. It first clarifies a few points concerning the structure and construction of Mumford-Tate domains before presenting a computationally effective procedure to determine the components in terms of Lie algebra representations and Weyl groups. It then shows that the normalizers of M in G are the groups stabilizing the Noether-Lefschetz locus. It also discusses the decomposition of Noether-Lefschetz loci into Hodge orientations, Weyl groups and permutations of Hodge orientations, and Galois groups and fields of definition. The results demonstrate that Mumford-Tate groups built up from well-understood real² factors are one source of easily described examples of Mumford-Tate domains.

This chapter deals with the Mumford-Tate group of a variation of Hodge structure (VHS). It begins by presenting a definition of VHS, which consists of a connected complex manifold and a locally liftable, holomorphic mapping that is an integral manifold of the canonical differential ideal. The moduli space of Γ‎-equivalence classes of polarized Hodge structures is also considered, along with a generic point for the VHS and the monodromy group of the VHS. Associated to a VHS is its Mumford-Tate group. The chapter proceeds by discussing the structure theorem for VHS, where S is a quasi-projective algebraic variety, referred to as global variations of Hodge structure. It concludes by describing an application of Mumford-Tate groups, along with the Noether-Lefschetz locus.

This chapter provides an introduction to the basic definitions and properties of Mumford-Tate groups in both the case of Hodge structures and of mixed Hodge structures. Hodge structures of weight n are sometimes called pure Hodge structures, and the term “Hodge structure” then refers to a direct sum of pure Hodge structures. The chapter presents three definitions of a Hodge structure of weight n, given in historical order. In the first definition, a Hodge structure of weight n is given by a Hodge decomposition; in the second, it is given by a Hodge filtration; in the third, it is given by a homomorphism of ℝ-algebraic groups. In the first two definitions, n is assumed to be positive and the p,q's in the definitions are non-negative. In the third definition, n and p,q are arbitrary. For the third definition, the Deligne torus integers are used.

This chapter describes Hodge structures with a high degree of symmetry, and specifically complex multiplication Hodge structures or CM Hodge structures. It broadens the notion of CM type by defining an n-orientation of a totally imaginary number field and constructs a precise correspondence between these and certain important kinds of CM Hodge structures. In the classical case of weight n = 1, the abelian variety associated to a CM type is recovered. The notion of the Kubota rank and reflex field associated to a CM Hodge structure is then generalized to the totally imaginary number field setting. When the Kubota rank is maximal, the CM Hodge structure is non-degenerate. The discussion also covers oriented imaginary number fields, Hodge structures with special endomorphisms, polarization and Mumford-Tate groups, and the Mumford-Tate group in the Galois case.