A tautological system, introduced in , 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 , and was verified for the case of projective homogeneous space under an assumption. In this paper, we prove this conjecture in full generality. By means of the Riemann–Hilbert correspondence and Fourier transforms, we also generalize the rank formula to an arbitrary projective manifold with a group action.
"Period integrals and the Riemann–Hilbert correspondence." J. Differential Geom. 104 (2) 325 - 369, October 2016. https://doi.org/10.4310/jdg/1476367060