Topological equisingularity for isolated complete intersection singularities. (English) Zbl 0751.14005

Parameswaran, A. J.

Compos. Math. 80, No. 3, 323-336 (1991).

Let \(X,x\) be a germ of an analytic variety (over the complex numbers) which is a complete intersection isolated singularity. The author associates to \(X,x\) a sequence \(\mu^*=(\mu_ 0,\mu_ 1,\ldots)\) of numerical invariants by taking \(\mu_ i\) to be the minimal value of the Milnor numbers \(\mu(X_ i,x_ i)\) for all deformations \((X_ i,x_ i)\to(S_ i,s_ i)\) of \(X,x\) with \(\dim S_ i=i\). One has \(\mu_ 0=\mu(X,x)\) and \(\mu_ i=0\) if \(i\) is bigger than the embedding codimension of \(X,x\). On the other hand the author defines the topological type of \(X,x\) as the class of homeomorphism of any sequence of germs \((X,x)=(X_ 0,x)\subset(X_ 1,x)\subset\cdots\subset(X_ k,x)\) where \(k\) is the embedding codimension of \(X,x\), for each \(i\), \(\mu(X_ i,x)=\mu_ i(X,x)\) and then the homeomorphism class does not depend on the \(X_ i\).
The main result in the paper says that a \(\mu^*\)-constant family of isolated complete intersection singularities of dimension different from two is topologically equisingular. A sufficient condition for the members of the family to have isomorphic monodromy fibrations is also given.

Reviewer: E.Casas-Alvero (Barcelona)

MSC:

14E15	Global theory and resolution of singularities (algebro-geometric aspects)
14B05	Singularities in algebraic geometry
32S40	Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
14M10	Complete intersections
32B10	Germs of analytic sets, local parametrization

Keywords:

equisingularity; topological type of singularity; complete intersection isolated singularity; Milnor numbers; isomorphic monodromy fibrations

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