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.

A tautological system, introduced in \cite{LSY}\cite{LY}, arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold X , equipped with a suitable Lie group action. In this article, we introduce two formulas -- one purely algebraic, the other geometric -- to compute the rank of the solution sheaf of such a system for CY hypersurfaces in a generalized flag variety. The algebraic version gives the local solution space as a Lie algebra homology group, while the geometric one as the middle de Rham cohomology of the complement of a hyperplane section in X . We use both formulas to find certain degenerate points for which the rank of the solution sheaf becomes 1. These rank 1 points appear to be good candidates for the so-called large complex structure limits in mirror symmetry. The formulas are also used to prove a conjecture of Hosono, Lian and Yau on the completeness of the extended GKZ system when X is $\P^n$.

A tautological system, introduced in [16][17], arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold X , equipped with a suitable Lie group action. A geometric formula for the holonomic rank of such a system was conjectured in [4], and was verified for the case of projective homogeneous space under an assumption. In this paper, we prove this conjecture in full generality. By means of the Riemann-Hilbert correspondence and Fourier transforms, we also generalize the rank formula to an arbitrary projective manifold with a group action.

A CY bundle on a compact complex manifold X was a crucial ingredient in constructing differential systems for period integrals in \cite{LY}, by lifting line bundles from the base X to the total space. A question was therefore raised as to whether there exists such a bundle that supports the liftings of all line bundles from X , simultaneously. This was a key step for giving a uniform construction of differential systems for arbitrary complete intersections in X . In this paper, we answer the existence question in the affirmative if X is assumed to be K\"ahler, and also in general if the Picard group of X is assumed to be free. Furthermore, we prove a rigidity property of CY bundles if the principal group is an algebraic torus, showing that such a CY bundle is essentially determined by its character map.

We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system , arising from the universal family of Calabi-Yau hypersurfaces $Y_a$ in a G-variety $X$ of dimension $n$. First, we construct a natural topological correspondence between relative cycles in $H_n (X − Y_a, \cup D − Y_a)$ bounded by the union of G-invariant divisors $\cup D$ in $X$ to the solution sheaf of $\tau$, in the form of chain integrals. Applying this to a toric variety with torus action, we show that in addition to the period integrals over cycles in $Y_a$, the new chain integrals generate the full solution sheaf of a GKZ system. This extends an earlier result for hypersurfaces in a projective homogeneous variety, whereby the chains are cycles [3, 7]. In light of this result, the mixed Hodge structure of the solution sheaf is now seen as the MHS of $H_n(X − Ya, \cup D − Y_a)$. In addition, we generalize the result on chain integral solutions to the case of general type hypersurfaces. This chain integral correspondence can also be seen as the Riemann-Hilbert correspondence in one homological degree. Finally, we consider interesting cases in which the chain integral correspondence possibly fails to be bijective.