Gauss-Manin connection of hyperelliptic polynomials
Let \(h:=y^2+P(x)\) be a hyperelliptic polynomial. In this very elegantly written paper, the author shows that the bifurcation set (where the topology of the fibers \(F_t:=h^{-1}(t)\) of \(h\) changes) coincides with the set \(C\) of critical values of \(h\). He provides an explicit expression of the Gauss-Manin connection associated to \(h\) and deduces an astonishing relation between the eigenvalues of the residue of this connection at critical values \(c\) and the spectra of the associated singular fiber \(F_c\). He computes an explicit basis of the de Rham spaces \(H^1(F_t)\) in terms of rational forms. He also shows that the linear monodromy operator at infinity of \(h\) is diagonalizable and deduces that the Riemann-Hilbert problem (whether every representation of \(\pi_1(P_1({\mathbb C})\backslash C)\) is the monodromy representation of a system of meromorphic differential equations on \(P_1({\mathbb C})\)) is solvable for the monodromy of \(h\). Finally, he describes the \({\mathcal D}\)-module associated to \(h\) and shows that the family \({\mathcal F}:=\{[1],[x],[x^2],\dots, [x^{n-1}]\}\) gives a basis of the \({\mathbb C}[t]\) module \[ M_0={\Omega^2({\mathbb C}^2)\over dh\wedge d\Omega^0({\mathbb C}^2)} \] (Brieskorn lattice).
scientific article; zbMATH DE number 1569458
scientific article