On the Holonomic Rank Problem

Spencer Bloch An Huang Harvard University Bong H. Lian Brandeis University Vasudevan Srinivas Tata Institute for Fundamental Research Shing-Tung Yau Harvard University


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 $\P^n$.
Holonomic Rank Problem, Lie algebra homology group
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  title={On the Holonomic Rank Problem},
  author={Spencer Bloch, An Huang, Bong H. Lian, Vasudevan Srinivas, and Shing-Tung Yau},
Spencer Bloch, An Huang, Bong H. Lian, Vasudevan Srinivas, and Shing-Tung Yau. On the Holonomic Rank Problem. 2013. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160828143705436456477.
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