PERIOD MAP SHIMURA

I. Period Domains

The classifying space D for Hodge structures of given type (h^{p,q}) is a homogeneous space:

D = G(R) / K where G = Aut(H_Q, Q) and K = stabilizer of a reference Hodge filtration

D is an open subset of the flag variety parametrizing filtrations F^p of the correct dimensions. Points of D correspond to polarized Hodge structures.

II. The Period Map

For a family of smooth projective varieties f: X → S, the period map sends each fiber to its Hodge structure:

Φ: S → Γ\D, s ↦ (H^n(X_s), F^•, Q)

where Γ = Im(π_1(S) → GL(H_Z)) is the monodromy group. This is a holomorphic, horizontal map.

III. Griffiths Transversality

The infinitesimal period relation (Griffiths transversality) constrains the derivative of the period map:

dΦ: T_s S → Hom(F^p, H/F^{p-1}) for all p

Equivalently: ∇(F^p) ⊆ F^{p-1} ⊗ Ω^1_S

This is an integrability condition. The period map is tangent to a distribution on D defined by the horizontal tangent bundle.

IV. Shimura Varieties

A Shimura variety Sh(G,X) is a moduli space of Hodge structures with given Mumford-Tate group G. When the period map has a Shimura target:

Sh(G,X) = G(Q)\(X × G(A_f)) / K_f

These are algebraic varieties over number fields. The Langlands program connects their cohomology to automorphic forms.

◆ PERIOD MAP SHIMURA

I. Period Domains

II. The Period Map

III. Griffiths Transversality

IV. Shimura Varieties