Local Polarized Period Mappings
Assume now not just that each Xb is Kähler, but that there is a Kähler class that varies holomorphically in b. In other words, assume there is a class ω in H2(X, Z) 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: FpHk(Xb, C) is orthogonal to Fk − p + 1Hk(Xb, C) with respect to Q.
- Positive difiniteness: For all p + q = k, the restriction of to the primitive classes of type (p, q) 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.
Read more about this topic: Period Mapping
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