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.

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Period mappings were introduced in the sixties [G] to study variation of complex structures of families of algebraic varieties. The theory of tautological systems was introduced recently [LSY,LY] 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.