Period Mapping - Ehresmann's Theorem

Ehresmann's Theorem

Let f : X → B be a holomorphic submersive morphism. For a point b of B, we denote the fiber of f over b by X_b. 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. That is, f−1(U) is diffeomorphic to X₀ × U. In particular, the composite map

is a diffeomorphism. This diffeomorphism is not unique because it depends on the choice of trivialization. The trivialization is constructed from smooth paths in U, and it can be shown that the homotopy class of the diffeomorphism depends only on the choice of a homotopy class of paths from b to 0. In particular, if U is contractible, there is a well-defined diffeomorphism up to homotopy.

The diffeomorphism from X_b 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.