Limits of tangents and minimality of complex links
Let \((X,x_0)\) denote the germ at some point \(x_0\) of a reduced analytic space embedded into \({\mathbb C}^N\). Consider the analytic coarsest Whitney stratification \(W = \{ W_i \}_{ i \geq 0}\) of \(X\) at \(x_0\) such that \(W_0 = \{ x_0 \}\). Denote by \(lk^{\mathbb C} (X,x_0)\) (\(lk^{\mathbb C} (X,W_i)\), resp.) the link of \(X\) at \(x_0\) (the link of the stratum \(W_i\) of \(X\), resp.). Recall that \(lk^{\mathbb C} (X,x_0)\) measures, in some sense, the singularity of \(X\) at \(x_0\) and that \(lk^{\mathbb C} (X,W_i)\) is called acyclic whenever it has the homology of a point. We state the main result of the paper which generalizes another one by \textit{B. Teissier} [Astérisque 7-8 (1973), 285-362 (1974; Zbl 0295.14003)]. Let \(F:({\mathbb C}^N, x_0) \rightarrow ({\mathbb C}, 0)\) be a local parameter such that the restriction to X, \(F'\), has at most one isolated singularity at \(x_0\). Assume that the complex link \(lk^{\mathbb C} (X,W_i)\) is not acyclic for any \(i \neq 0\). Then we have the equivalences: (a) The Milnor fibre \(M(F', x_0)\) is homotopy equivalent to the complex link \(lk^{\mathbb C} (X, x_0)\). (b) The Milnor fibre \(M(F', x_0)\) has the same homology groups as \(lk^{\mathbb C} (X, x_0)\). (c) \(F\) is general with respect to \(X\) and \(W\) at \(x_0\). To conclude this review, note that the author also proves that the nonacyclicity assumption for the complex links is satisfied, for instance, by any space \(X\) with isolated singularity at \(x_0\) and by any complete intersection with at most one-dimensional singularities.
scientific article; zbMATH DE number 1929319