Period Mapping - Global Period Mappings

Global Period Mappings

Focusing only on local period mappings ignores the information present in the topology of the base space B. The global period mappings are constructed so that this information is still available. The difficulty in constructing global period mappings comes from the monodromy of B: There is no longer a unique homotopy class of diffeomorphisms relating the fibers X_b and X₀. Instead, distinct homotopy classes of paths in B induce possibly distinct homotopy classes of diffeomorphisms and therefore possibly distinct isomorphisms of cohomology groups. Consequently there is no longer a well-defined flag for each fiber. Instead, the flag is defined only up to the action of the fundamental group.

In the unpolarized case, define the monodromy group Γ to be the subgroup of GL(Hk(X₀, Z)) consisting of all automorphisms induced by a homotopy class of curves in B as above. The flag variety is a quotient of a Lie group by a parabolic subgroup, and the monodromy group is an arithmetic subgroup of the Lie group. The global unpolarized period domain is the quotient of the local unpolarized period domain by the action of Γ (it is thus a collection of double cosets). In the polarized case, the elements of the monodromy group are required to also preserve the bilinear form Q, and the global polarized period domain is constructed as a quotient by Γ in the same way. In both cases, the period mapping takes a point of B to the class of the Hodge filtration on X_b.