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Published online by Cambridge University Press: 11 November 2009
Abstract
Let (X, x) be an isolated complete intersection singularity and let f: (X, x) → (ℂ, 0) be the germ of an analytic function with an isolated singularity at x. An important topological invariant in this situation is the Picard-Lefschetz monodromy operator associated to f. We give a survey on what is known about this operator. In particular, we review methods of computation of the monodromy and its eigenvalues (zeta function), results on the Jordan normal form of it, definition and properties of the spectrum, and the relation between the monodromy and the topology of the singularity.
Introduction
The word ‘monodromy’ comes from the greek word μονο – δρομψ and means something like ‘uniformly running’ or ‘uniquely running’. According to [99, 3.4.4], it was first used by B. Riemann [135]. It arose in keeping track of the solutions of the hypergeometric differential equation going once around a singular point on a closed path (cf. [30]). The group of linear substitutions which the solutions are subject to after this process is called the monodromy group.
Since then, monodromy groups have played a substantial rôle in many areas of mathematics. As is indicated on the webside ‘www.monodromy.com’ of N.M. Katz, there are several incarnations, classical and l-adic, local and global, arithmetic and geometric. Here we concentrate on the classical local geometric monodromy in singularity theory. More precisely we focus on the monodromy operator of an isolated hypersurface or complete intersection singularity.
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