On the holonomic rank problem

A tautological system, introduced in [17, 18], 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 algebrahomology 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 Pn [12].