Period Matrices
The Hodge filtration can be expressed in coordinates using period matrices. Choose a basis δ1, ..., δr for the torsion-free part of the kth integral homology group Hk(X, Z). Fix p and q with p + q = k, and choose a basis ω1, ..., ωs 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.
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