I am considering the following function of z on the Reimann sphere

\begin{equation} J(z) = \int_\Delta (L_0+z L_1)^a D^b d^nx \end{equation}

where $\Delta \in H_n(CP^n-\{L(x)=0\},S)$,$S$ being the standard simplex,$L(x) = L_(x)+zL_1(x)$, and $L_i(x),D(x)$ are polynomials of $x=(x_1,...,x_n)$ with integer coefficients.

Lemma1: 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 Arnold-Varchenko-Gusein-Zade). One may be tempted to conjecture that J(z) can be included into a bundle as follows

\begin{equation} dI(z)/dz = \sum \frac{A_{I,a}}{z-z_{I,a}} I \end{equation} 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:

1. 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" ch2)

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.

Case 1. in particular is treated in the works of Moser-Barkatou-Pfluger-... where the conditions necessary 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?

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.