## Journal of Differential Geometry

### 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.

#### Article information

Source
J. Differential Geom., Volume 97, Number 1 (2014), 11-35.

Dates
First available in Project Euclid: 9 July 2014

https://projecteuclid.org/euclid.jdg/1404912101

Digital Object Identifier
doi:10.4310/jdg/1404912101

Mathematical Reviews number (MathSciNet)
MR3229048

Zentralblatt MATH identifier
1318.32027

#### Citation

Bloch, Spencer; Huang, An; Lian, Bong H.; Srinivas, Vasudevan; Yau, Shing-Tung. On the holonomic rank problem. J. Differential Geom. 97 (2014), no. 1, 11--35. doi:10.4310/jdg/1404912101. https://projecteuclid.org/euclid.jdg/1404912101