On the holonomic rank problem

Abstract

A tautological system, introduced in Period Integrals and Tautological Systems and Period Integrals of CY and General Type Complete Intersections, 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 $\mathbb{P}^n$, GKZ-generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces.