Holonomic systems for period mappings

Period mappings were introduced in the sixties [4] to study variation of complex structures of families of algebraic varieties. The theory of tautological systems was introduced recently [7] and [8] to understand period integrals of algebraic manifolds. In this paper, we give an explicit construction of a tautological system for each component of a period mapping. We also show that the D-module associated with the tautological system gives rise to many interesting vanishing conditions for period integrals at certain special points of the parameter space.