Brieskorn and the monodromy

Abstract

Brieskorn’s paper “Die Monodromie der isolierten Singularit aten von Hyperfl aschen,” pub- lished in 1970 in Manuscripta Mathematica, gave a new insight to the theory of monodromy and Gaus-Manin connections. The paper, written in the framework of isolated hypersurface singularities, has been generalized for isolated complete intersection singularities by G.-M. Greuel in 1975 [10]. In the following times and also more recently, a long list of authors, among them P. Deligne [7], W. Ebeling [8], H. Hamm [12], Lˆe D. T. [20], B. Malgrange [24], D.Siersma [37] etc. provided generalizations and developments of the monodromy theory. The regularity of the Gaus-Manin connection, proved by Brieskorn in the framework of isolated hypersurface singularities has been proved and developed in various situations by many authors, among them G.-M. Greuel [10], C. Hertling [15], F. Pham [28], K. Saito [29], M. Saito [30], J. Scherk and J.H.M. Steenbrink [31], M. Schulze [32], A. Varchenko [38], etc. There are many surveys concerning the various aspects of monodromy and including developments of the theory. In particular, Ebeling’s survey [8] shows very well the importance of Brieskorn’s article as well as developments and generalisations of the Brieskorn’s results. Siersma’s survey [37] deals with the non-isolated case, and presents new results in this framework. The present paper, based on ideas of the second author [34, 35, 36], does not pretend any originality. It is not devoted to specialists, but to “beginners”. The aim of the paper is to introduce monodromy theory and provide some elementary view about the Brieskorn paper. Our aim is not to replace the reading of this very important Brieskorn article, but hopefully to encourage one to read it.