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arXiv · 1302.4481

On the Holonomic Rank Problem

Spencer Bloch·An Huang·Bong H. Lian·Vasudevan Srinivas·Shing-Tung Yau
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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 $\P^n$.

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Spencer Bloch, An Huang, Bong H. Lian, Vasudevan Srinivas, Shing-Tung Yau. 2013-02-18. On the Holonomic Rank Problem. https://arxiv.org/abs/1302.4481

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