Gauss-Manin connection of hyperelliptic polynomials
type: Item
instance of: scholarly article
title: Gauss-Manin connection of hyperelliptic polynomials
zbMATH Open document ID: 0997.32023
DOI: 10.1016/S0007-4497(00)01063-0
author: Gilles Bailly-Maitre
published in: Bulletin des Sciences Mathématiques
publication date: 2002-11-07T00:00:00Z
review text: 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).
reviewed by: Joël Merker
Mathematics Subject Classification ID: 32S20
Mathematics Subject Classification ID: 58A12
Mathematics Subject Classification ID: 34M50
zbMATH DE Number: 1569458
zbMATH Keywords: hyperelliptic polynomial
zbMATH Keywords: bifurcation set
zbMATH Keywords: topology of polynomial fibers
zbMATH Keywords: Milnor number
zbMATH Keywords: monodromy representation
zbMATH Keywords: Riemann-Hilbert problem
zbMATH Keywords: Brieskorn lattice
zbMATH Keywords: Gauss-Manin connection
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1569458