Period integrals and the Riemann-Hilbert correspondence (Q342699)

A tautological system, introduced in [\textit{B. H. Lian} et al., J. Eur. Math. Soc. (JEMS) 15, No. 4, 1457--1483 (2013; Zbl 1272.14033)] and [\textit{B. H. Lian} and \textit{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 \(X\), equipped with a suitable Lie group action. A geometric formula for the holonomic rank of such a system was conjectured in [\textit{S. Bloch} et al., J. Differ. Geom. 97, No. 1, 11--35 (2014; Zbl 1318.32027)], and was verified for the case of projective homogeneous space under an assumption. In this paper, the authors prove this conjecture in full generality. By means of the Riemann-Hilbert correspondence and Fourier transforms, they also generalize the rank formula to an arbitrary projective manifold with a group action.