Critical points of functions on analytic varieties. (English) Zbl 0639.32008

Bruce, J. W.; Roberts, R. M.

Topology 27, No. 1, 57-90 (1988).

Let (X,0) be the germ of an analytic subvariety in \(({\mathbb{C}}^ n,0)\). For a holomorphic function f on \(({\mathbb{C}}^ n,0)\) the authors define the Milnor number \(\mu_ X(f)\) of f with respect to X as the dimensions (over \({\mathbb{C}})\) of the quotient of \({\mathcal O}_{{\mathbb{C}}^ n,0}\) by the ideal generated by the function \(\partial f\), where \(\partial\) is a vector field on \(({\mathbb{C}}^ n,0)\) parallel to (X,0). They show that this number has many properties analogous to the usual Milnor number - which is the special case \(X=\emptyset\). In particular they show that the codimension of the orbit of f in \({\mathcal O}_{{\mathbb{C}}^ n,0}\) under the group of automorphisms of \(({\mathbb{C}}^ n,0)\) preserving (X,0) is equal to \(\mu_ X(f)\) if this number is finite, and they generalize the property that for \(X=\emptyset\) the number of critical points of a morsification of f equals \(\mu_ X(f)\). The concepts and results developed in the paper are applied to various classes of analytical varieties (X,0), for example to free divisors (i.e. reduced hypersurfaces whose sheaf of vector fields is free over \({\mathcal O}_{{\mathbb{C}}^ n,0})\), complete intersections and quotient singularities.

Reviewer: H.Knörrer

MSC:

32Sxx	Complex singularities
32S30	Deformations of complex singularities; vanishing cycles
32S60	Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
14B07	Deformations of singularities
58C25	Differentiable maps on manifolds
58K99	Theory of singularities and catastrophe theory

Keywords:

stratification; holomorphic function; Milnor number; free divisors; complete intersections; quotient singularities