Period Mapping - Local Unpolarized Period Mappings

Local Unpolarized Period Mappings

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 FpHk(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 FpHk(X_b, C) is independent of b. The period map is the map

where F is the flag variety of chains of subspaces of dimensions b_p,k for all p, that sends

Because X_b is a Kähler manifold, the Hodge filtration satisfies the Hodge–Riemann bilinear relations. These imply that

Not all flags of subspaces satisfy this condition. The subset of the flag variety satisfying this condition is called the unpolarized local period domain and is denoted . is an open subset of the flag variety F.