b. In other words, assume there is a class ω in such that for every b, the restriction ωb of ω to Xb is a Kähler class. ωb determines a bilinear form Q on Hk(Xb, C) by the rule This form varies holomorphically in b, and consequently the image of the period mapping satisfies additional constraints which again come from the Hodge–Riemann bilinear relations. These are: Orthogonality: is orthogonal to with respect to Q. Positive definiteness: For all , the restriction of to the primitive classes of type is positive definite. The polarized local period domain is the subset of the unpolarized local period domain whose flags satisfy these additional conditions. The first condition is a closed condition, and the second is an open condition, and consequently the polarized local period domain is a locally closed subset of the unpolarized local period domain and of the flag variety F. The period mapping is defined in the same way as before. The polarized local period domain and the polarized period mapping are still denoted and , respectively. 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 Xb 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