Period mapping

In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures.

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.

Assume that f is proper and that X₀ is a Kähler variety. The Kähler condition is open, so after possibly shrinking U, X_b 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 X_b. These isomorphisms of cohomology groups will not in general preserve the Hodge structures of X₀ and X_b because they are induced by diffeomorphisms, not biholomorphisms. Let F^pH^k(X_b, C) denote the pth step of the Hodge filtration. The Hodge numbers of X_b are the same as those of X₀, so the number b_p,k = dim F^pH^k(X_b, C) is independent of b. The period map is the map

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