2013/02/18 by Spencer Bloch, Bloch, Spencer, An Huang +7 · 2 citations
Mathematics · #14D07 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.1302.4481
openalex publication_date 2013/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A tautological system, introduced in \citeLSY\citeLY, 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 ¶n.