Fix a point 0 in B. Ehresmann's theorem guarantees that there is a small open neighborhood U around 0 in which f becomes a fiber bundle.
The diffeomorphism from Xb to X0 induces an isomorphism of cohomology groups and since homotopic maps induce identical maps on cohomology, this isomorphism depends only on the homotopy class of the path from b to 0.
These isomorphisms of cohomology groups will not in general preserve the Hodge structures of X0 and Xb because they are induced by diffeomorphisms, not biholomorphisms.
The subset of the flag variety satisfying this condition is called the unpolarized local period domain and is denoted
ωb determines a bilinear form Q on Hk(Xb, C) by the rule This form varies holomorphically in b, and consequently the image of the period mapping satisfies additional constraints which again come from the Hodge–Riemann bilinear relations.
Focusing only on local period mappings ignores the information present in the topology of the base space B.
The difficulty in constructing global period mappings comes from the monodromy of B: There is no longer a unique homotopy class of diffeomorphisms relating the fibers Xb and X0.
In the unpolarized case, define the monodromy group Γ to be the subgroup of GL(Hk(X0, Z)) consisting of all automorphisms induced by a homotopy class of curves in B as above.
In both cases, the period mapping takes a point of B to the class of the Hodge filtration on Xb.
Choose a basis δ1, ..., δr for the torsion-free part of the kth integral homology group Hk(X, Z).
A period matrix is equivalent to Ω if it can be written as AΩΛ for some choice of A and Λ.
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 ω = dx/y.
Each half of this curve connects the points 1 and λ on the two sheets of the Riemann surface.
The bilinear form √−1Q is positive definite because locally, we can always write ω as f dz, hence By Poincaré duality, γ and δ correspond to cohomology classes γ* and δ* which together are a basis for H1(X0, Z).
The coefficients are given by evaluating ω with respect to the dual basis elements γ and δ: When we rewrite the positive definiteness of Q in these terms, we have Since γ* and δ* are integral, they do not change under conjugation.