Period Mapping - Period Matrices

Period Matrices

The Hodge filtration can be expressed in coordinates using period matrices. Choose a basis δ₁, ..., δ_r for the torsion-free part of the kth integral homology group H_k(X, Z). Fix p and q with p + q = k, and choose a basis ω₁, ..., ω_s for the harmonic forms of type (p, q). The period matrix of X₀ with respect to these bases is the matrix

The entries of the period matrix depend on the choice of basis and on the complex structure. The δs can be varied by a choice of a matrix Λ in SL(r, Z), and the ωs can be varied by a choice of a matrix A in GL(s, C). A period matrix is equivalent to Ω if it can be written as AΩΛ for some choice of A and Λ.