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The monodromy of a series of hypersurface singularities. (English) Zbl 0723.32015
Let \(\{f=0,0\}\) be a germ of a hypersurface singularity in \(({\mathbb{C}}^{n+1},0)\) with a 1-dimensional singular set \(\Sigma\). If x is a generic linear form, a little deformation \(f+\epsilon x^ N\) (\(\epsilon\) little, N big) has an isolated singularity. Let S be a local irreducibe component of \(\Sigma\) at 0. Along S-\(\{\) \(0\}\), f can be viewed as a \(\mu\)-constant deformation of the transversal section (which has an isolated singularity). These singularities have a monochromy (called horizontal) and the local system over S-\(\{\) \(0\}\) defines another monodromy (called vertical).
The main theorem of this paper relates the characteristic polynomials of the monodromies of f, \(f+\epsilon x^ N\), the vertical and the horizontal monodromies. The methods, polar curves and carroussel, are essentially topological.
Almost simultaneously, M. Saito has proved in this situation the Steenbrink conjecture [M. Saito, Math. Ann. 289, 703-716 (1991)], which relates the spectra of f and of \(f+\epsilon x^ N\). The monodromy is characterized by the values mod \({\mathbb{Z}}\) of the spectrum. This better result is proved by the Saito theory of mixed Hodge modules.

MSC:
32S25 Complex surface and hypersurface singularities
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
32S30 Deformations of complex singularities; vanishing cycles
32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
14B05 Singularities in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
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