Open Access
May 2014 On the holonomic rank problem
Spencer Bloch, An Huang, Bong H. Lian, Vasudevan Srinivas, Shing-Tung Yau
J. Differential Geom. 97(1): 11-35 (May 2014). DOI: 10.4310/jdg/1404912101


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.


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Spencer Bloch. An Huang. Bong H. Lian. Vasudevan Srinivas. Shing-Tung Yau. "On the holonomic rank problem." J. Differential Geom. 97 (1) 11 - 35, May 2014.


Published: May 2014
First available in Project Euclid: 9 July 2014

zbMATH: 1318.32027
MathSciNet: MR3229048
Digital Object Identifier: 10.4310/jdg/1404912101

Rights: Copyright © 2014 Lehigh University

Vol.97 • No. 1 • May 2014
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