Topological finite-determinacy of functions with non-isolated singularities (Q1884663)

This paper presents a generalization of a theorem of A. N. Varchenko, related to a result of R. Thom on finite determination of germs of functions. Varchenko's theorem, in the setting of complex analytic functions, says that if \(T\) is an irreducible algebraic subset of the space of jets \(J^r(C^n,C)_0\), then for suitable \(s\geq r\) there is a proper algebraic subset \(A\subset(\pi_r^s)^{-1}(T)\) such that two germs of functions \(f\) and \(g\) with a common \(s\)-jet, which is not in \(A\), have the same topological type. Moreover, there is a partition of \(J^r(C^n,C)_0\) into disjoint constructible subsets \(U^r_0,\dots,U^r_{k(r)}\), invariant under the action of \({\mathcal D}\) (the group of germs of biholomorphisms at the origin), such that if two germs \(f_1\) and \(f_2\) have the same \(r\)-jet in \(U_j\) \(j>0\), then they are of the same topological type; the codimension of \(U_0^r\) tends to infinity as \(r\) increases. This result is not helpful for germs of functions with non-isolated singularities at 0, because they correspond to the stratum \(U^r_0\). This paper deals with results that lift that restriction. In his generalization, the author does not work with the ring of germs of functions at the origin, but rather with a coherent sheaf \({\mathcal I}\) of ideals defined on a suitable neighborhood \(W\) of the origin. Rather than working with the usual space of jets, he introduces and uses spaces \(J^r(W,{\mathcal I})\) and \(J^\infty(W,{\mathcal I})\). After introducing and studying a number of concepts (finite determinacy of closed analytic subsets of \(J^\infty(W,{\mathcal I})\), suitable discriminants, filtrations, etc.), he states and proves generalizations of both parts of Varchenko's theorem, where functions with non-isolated singularities do not play a special role. These are proposition 9 and the main theorem (in section 4). The statements are too long and technical to be repeated here. But let us mention that the notion of \({\mathcal D}\)-invariance must be replaced by that of flow-invariant set. An application of these results is a proof of the existence of generic topological types of germs from \({\mathcal I}_0\) varying in a family parametrized by an irreducible analytic set. The author works both over the complex and real numbers, in the latter case the requirement is that the objects involved admit a nice complexification.