On the Holonomic Rank Problem

TL;DR

This paper introduces algebraic and geometric formulas to compute the rank of tautological systems governing period integrals in complex geometry, identifying special points relevant to mirror symmetry and confirming a conjecture for projective spaces.

Contribution

It provides new algebraic and geometric methods to compute solution sheaf ranks of tautological systems, advancing understanding in mirror symmetry and hypergeometric systems.

Findings

Derived formulas for solution sheaf rank in CY hypersurfaces

Identified rank 1 points as potential large complex structure limits

Proved a conjecture on the completeness of the extended GKZ system for projective spaces

Abstract

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 $\P^n$.