  We develop a global Poincar\'e residue formula to study period integrals of
families of complex manifolds. For any compact complex manifold $X$ equipped
with a linear system $V^*$ of generically smooth CY hypersurfaces, the formula
expresses period integrals in terms of a canonical global meromorphic top form
on $X$. Two important ingredients of our construction are the notion of a CY
principal bundle, and a classification of such rank one bundles. We also
generalize our construction to CY and general type complete intersections. When
$X$ is an algebraic manifold having a sufficiently large automorphism group $G$
and $V^*$ is a linear representation of $G$, we construct a holonomic D-module
that governs the period integrals. The construction is based in part on the
theory of tautological systems we have developed in the paper \cite{LSY1},
joint with R. Song. The approach allows us to explicitly describe a
Picard-Fuchs type system for complete intersection varieties of general types,
as well as CY, in any Fano variety, and in a homogeneous space in particular.
In addition, the approach provides a new perspective of old examples such as CY
complete intersections in a toric variety or partial flag variety.
