  A tautological system, introduced in [16][17], 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. A geometric formula for the holonomic rank of such a
system was conjectured in [4], 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.
