Period map

Summary

The period map of a family of compact Kähler manifolds records how the Hodge structure on the cohomology of the fibres moves. All fibres have the same cohomology group, and only the position of the Hodge filtration inside it changes. So the period map goes from the base of the family to a space of flags, a period domain. Griffiths showed that it is holomorphic and satisfies a transversality condition. Classical periods of elliptic curves and abelian integrals are the simplest case.