An Introduction to Hodge Structures and Shimura Varieties

In this chapter we recall the general facts about Hodge structures and Shimura varieties, which are needed in the sequel. We will explain that a Shimura datum consisting of a Q-reductive group G and a homomorphism h : S → G ^R satisfying certain conditions allows the construction of a Hermitian symmetric domain D. We will also give a definition of complex multiplication (CM), give a criterion for complex multiplication and discuss some conjectures concerning complex multiplication.

Shimura varieties and complex multiplication are closely related. One can construct a variation of Hodge structures on a Hermitian symmetric domain obtained from a Shimura datum. This variation of Hodge structures yields Hodge structures with complex multiplication over a dense set of points. Due to the André-Oort conjecture, one assumes that every variation of Hodge structures which contains infinitely many Hodge structures with complex multiplication is of this kind.

In the first two sections we recall the basic definitions of Hodge theory and consider polarized integral Hodge structures of type (1, 0), (0, 1), which correspond to isomorphism classes of polarized abelian varieties with symplectic basis by Riemann’s theorem. We define Shimura data and construct Hermitian symmetric domains by using Shimura data in Section 1.3 and Section 1.4 respectively. The construction of Shimura varieties from the Hermitian symmetric domains obtained by Shimura data is sketched in Section 1.5.