On a conjecture of Huang-Lian-Yau-Yu (Q6628890)

For an \(n\)-dimensional, smooth, projective variety \(X\) with an action of a connected algebraic group \(G\), let \(r\) ample invertible sheaves on it be given. For \(r=1\), the tautological system was introduced by \textit{B. H. Lian} et al. [J. Eur. Math. Soc. (JEMS) 15, No. 4, 1457--1483 (2013; Zbl 1272.14033)] with the aim to study periods of Calabi-Yau hypersurfaces in \(X\). Tautological systems specialize to the Gelfand-Kapranov-Zelevinsky A-hypergeometric systems when \(G\) is a torus. The authors verify a formula on the solution rank of the tautological system arising from ample complete intersections in a projective homogeneous space of a semisimple group conjectured by Huang-Lian-Yau-Yu. As an application, they prove the existence of the rank one point for such a system, where mirror symmetry is expected.