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, 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. 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.
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