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Discriminants and liftable vector fields. (English) Zbl 0821.58007

Let \(f : (M,S) \to (N,0)\) be a complex analytic map-germ, where \(M\) and \(N\) are complex analytic manifolds and \(S\) is finite. Denote by \(\Sigma(f)\) the critical set and by \(\Delta(f) = F(\Sigma(f))\) the discriminant set of \(f\). The authors pose the question of which map-germs \(f\) are determined by their discriminant set up to \(\mathcal A\)- or \(\mathcal R\)- equivalence. In particular, they prove that there are at most finitely many \(\mathcal A\)-classes of \(\mathcal A\)-finite analytic map-germs with target dimension greater than one with given source dimension and given discriminant set.
It should be noted that from an earlier work [the second and third author, Math. Z. 190, 163-205 (1985; Zbl 0593.58014)] it follows that there is only one \(\mathcal R\)-class (and, hence, one \(\mathcal A\)-class) of \(\mathcal A\)-finite map-germs with given discriminant set in the cases \(\dim N > 2\) or \(\dim M = 1\), \(\dim N = 2\). Thus, in the paper under review the remaining case \(\dim M \geq \dim N = 2\) has been analyzed completely. In conclusion the methods developed by the authors are applied to the problem of classification of mappings from the plane to the plane. In fact, it turns out that the \(\mathcal A\)-classification is essentially reduced to the \(\mathcal K\)-classification of singular points of plane algebraic curves.

MSC:

58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
57R70 Critical points and critical submanifolds in differential topology
32C07 Real-analytic sets, complex Nash functions
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32C20 Normal analytic spaces
58A07 Real-analytic and Nash manifolds
58K35 Catastrophe theory

Citations:

Zbl 0593.58014
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