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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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