On the holonomic rank problem.

*(English)*Zbl 1318.32027A tautological system, as introduced previously by B. H. Lian et al. [J. Eur. Math. Soc. (JEMS) 15, No. 4, 1457–1483 (2013; Zbl 1272.14033)] and B. H. Lian and S.-T. Yau [Invent. Math. 191, No. 1, 35–89 (2013; Zbl 1276.32004)], arises as a regular holonomic system of partial differential equations that governs the period integrals of a family of complete intersections in a complex manifold \(X\) that is endowed with a suitable Lie group action.

In the present paper the authors introduce two formulas (one purely algebraic, the other one geometric) to compute the rank of the solution sheaf of such a system for CY hypersurfaces in a generalized flag variety. The algebraic formula expresses the solution space at any given point (singular or not) as the dual of a certain Lie algebra homology with coefficient in the coordinate ring of \(X\). The geometric formula uses the algebraic result to identify the solution space with the middle de Rham cohomology of the complement of the same CY hyperplane section in X. Based on much numerical evidence, it is conjectured that the geometric result holds for an arbitrary homogeneous variety. Their proof is valid for most familiar cases (e.g., projective spaces, Grassmannians, quadrics, spinor varieties, maximal Lagrangian Grassmannians, two exceptional varieties, full flag varieties \(G/B\), and products of such).

Moreover, they 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 S. Hosono et al. [Commun. Math. Phys. 182, No. 3, 535–577 (1996; Zbl 0870.14028)] on the completeness of the extended GKZ system when \(X\) is \({\mathbb P}^n\).

In the present paper the authors introduce two formulas (one purely algebraic, the other one geometric) to compute the rank of the solution sheaf of such a system for CY hypersurfaces in a generalized flag variety. The algebraic formula expresses the solution space at any given point (singular or not) as the dual of a certain Lie algebra homology with coefficient in the coordinate ring of \(X\). The geometric formula uses the algebraic result to identify the solution space with the middle de Rham cohomology of the complement of the same CY hyperplane section in X. Based on much numerical evidence, it is conjectured that the geometric result holds for an arbitrary homogeneous variety. Their proof is valid for most familiar cases (e.g., projective spaces, Grassmannians, quadrics, spinor varieties, maximal Lagrangian Grassmannians, two exceptional varieties, full flag varieties \(G/B\), and products of such).

Moreover, they 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 S. Hosono et al. [Commun. Math. Phys. 182, No. 3, 535–577 (1996; Zbl 0870.14028)] on the completeness of the extended GKZ system when \(X\) is \({\mathbb P}^n\).

Reviewer: Anna Fino (Torino)