quotient by Γ in the same way. In both cases, the period mapping takes a point of B to the class of the Hodge filtration on Xb. Properties Griffiths proved that the period map is holomorphic. His transversality theorem limits the range of the period map. 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 . Fix p and q with , and choose a basis ω1, ..., ωs for the harmonic forms of type . 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 , and the ωs can be varied by a choice of a matrix A in . A period matrix is equivalent to Ω if it can be written as AΩΛ for some choice of A and Λ. The case of elliptic curves Consider the family of elliptic curves where λ is any complex number not equal to zero or one. The Hodge filtration on the first cohomology group of a curve has two steps, F0 and F1. However, F0 is the entire cohomology group, so the only interesting term of the filtration is F1, which is H1,0, the space of holomorphic harmonic 1-forms. H1,0 is one-dimensional because the curve is elliptic, and for all λ, it is spanned by the differential form . To find explicit representatives of the homology group of the curve, note that the curve can be represented as the graph of the multivalued function on the Riemann sphere. The branch points of this function are at zero, one, λ, and infinity. Make two branch cuts, one running from zero to one and the other running from λ to infinity. These exhaust the branch points of the function, so they cut the multi-valued function into two single-valued sheets. Fix a small . On one of these sheets, trace the curve . For ε sufficiently small, this curve surrounds the branch cut and does not meet the branch cut . Now trace another curve δ(t) that begins in one sheet as for and continues in the other sheet as for . Each half of this curve connects the points