Construction of a Pfaff system of Fuchs type with three singular surfaces on a complex projective space (Q731558)

Let \(\overline M_ j=\{x;\;P_ j(x)=0\} (j=1,2,3)\) be irreducible algebraic varieties of co-dimension \(1\) on the \(n\)-dimensional complex projective space \(\mathbb{C}\mathbb{P}^ n\). Let \(d Y=\omega Y\) be the linear Pfaff system, where \(\omega=\sum_{j=1}^ 3 U_ j\frac{dP_ j(x)}{P_ j(x)}\) is a differential \(1\)-form, \(Y\) is a second-order square matrix, \(U_ j\) are constant square \(2\times 2\)-matrices, and \(P_ j(x)\) are irreducible homogeneous polynomials of degree \(p_ j\) in \(x=(x_ 1,\dots,x_{n+1})\). For an open set \(M=\mathbb{C}\mathbb{P}^ n-\bigcup_{j=1}^ 3\overline M_ j\), let \(\chi:\pi_ 1\left(x_{_ 0},M\right)\to GL(2,C)\) be a given homomorphism of the fundamental group \(\pi_ 1\left(x_{_ 0},M\right)\) into the general linear group of square non-singular second-order matrices. \(\chi\) takes the generators \(g_ j\) of the fundamental group \(\pi_ 1\left(x_{_ 0},M\right)\) to some monodromy matrices \(V_ j\). The authors construct a multidimensional Pfaff equation of Fuchs type for a given monodromy group on \(\mathbb{C}\mathbb{P}^ n\). They show that if the irreducible algebraic varieties \(\overline M_ j\) form a pencil of surfaces, i.e., \(P_ j=\alpha_ j R+\beta_ j Q\), then the change of variables \(z=\frac{Q(x)}{R(x)}\) reduces the Pfaff equation \(d Y=\omega Y\) to the ordinary linear differential equation \(dY=\left(\sum_{i=1}^ 3\frac{U_ i}{z+\alpha_ i/\beta_ i}\right)dz Y\) with Fuchs singularities on \(\mathbb{C}\mathbb{P}^ 1\).