I am considering the following function of $z$ on the Riemann sphere: $$ J(z) = \int_\Delta (L_0+z L_1)^a D^b d^nx $$ where
- $\Delta \in H_n\big(\Bbb{CP}^n\setminus\{L(x)=0\},S\big)$, $S$ being the standard simplex,
- $L(x) = L_0(x)+zL_1(x)$, and $L_i(x),D(x)$ are polynomials of $x=(x_1,...,x_n)$ with integer coefficients.
Lemma 1: the singularities of $J(z)$ are contained in the set where $L$ or its restrictions $L|_I$ to coordinate planes define singular projective hypersurfaces ( assuming that for the generic $z$ it is a non singular hypersurface). Denote these points $z_{I,a}$
It is known that $J(z)$ defines a flat vector bundle with regular singularities (see Arnol'd-Varchenko-Gusein-Zade). One may be tempted to conjecture that $J(z)$ can be included into a bundle as follows $$ \frac{dI(z)}{dz} = \sum \frac{A_{I,a}}{z-z_{I,a}} I $$ where $A_{I,a}$ are constant matrices, $J$ being one of the components of the vector function $I$.
Unfortunately, this does not follow from general principles because:
There can be non Fuchsian regular singularities, where the bundle equation has a pole of higher order, but the solution is still regular ( see Anosov Bolibrukh "The Riemann Hilbert problem" ch. 2)
There can be apparent singularities, where the equation is singular but the solution is not.
There is a large literature on each of these cases. In particular,
Case 1, in particular is treated in the works of Moser-Barkatou-Pfluger-... where the necessary conditions for regularity are studied.
Case 2, has very interesting connections to isomonodromic deformations, treated e.g. in the works of Vyugin, Gontsov ( see also Poberezhny, Eremenko, Cohen, Wolfart,....)
Question: do these complications arise in the case of the functions defined by the integral above?
There is evidence that they do not. The evidence comes from the existence of a holonomic $D$-module for universal deformation of the parameter space (where all the coefficients are deformed from integers). This comes from Gelfand-Kapranov-Zelevinsky theory. Then it is known that for generic polynomials, the $D$-module can be converted to the Gauss-Manin connection. Then our case can be obtained as a restriction of that GM connection to 1d subspace.
