Period mapping

In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures. 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 Xb. 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 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. Assume that f is proper and that X0 is a Kähler variety. The Kähler condition is open, so after possibly shrinking U, Xb is compact and Kähler for all b in U. After shrinking U further we may assume that it is contractible. Then there is a well-defined isomorphism between the cohomology groups of X0 and Xb. 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. Let FpHk(Xb, C) denote the pth step of the Hodge filtration. The Hodge numbers of Xb are the same as those of X0, so the number bp,k = dim FpHk(Xb, C) is independent of b. The period map is the map where F is the flag variety of chains of subspaces of dimensions bp,k for all p, that sends Because Xb is a Kähler manifold, the Hodge filtration satisfies the Hodge–Riemann bilinear relations. These imply that Not all flags of subspaces satisfy this condition.