Isosingular Loci and the Cartesian Product Structure of Complex Analytic Singularities

Let X be a (not necessarily reduced) complex analytic space, and let F be a germ of an analytic space. The locus of points q in X at which the germ Xq is complex analytically isomorphic to F is studied. If it is nonempty it is shown to be a locally closed submanifold of X, and X is locally a Cartesian product along this submanifold. This is used to define what amounts to a coarse partial ordering of singularities. This partial ordering is used to show that there is an essentially unique way to completely decompose an arbitrary reduced singularity as a cartesian product of lower dimensional singularities. This generalizes a result previously known only for irreducible singularities. 0. Introduction. Let X be a complex analytic space. For q E X, Xq will denote the germ of X at q. In this paper I will study the isosingular loci defined by Definition 0.1. Forp G X let lso{X,p) = {qEX\Xq = Xp). (¡a here and elsewhere will mean complex analytically isomorphic.) It will be shown that: Theorem 0.2. For any p E X, lso{X,p) is a {possibly 0-dimensional) complex submanifold of some open subset of X. Moreover, for any q E Iso(A", p) there is an open neighbornood U of q, and an analytic space Y such that U at Y X {U n lso{X,p)). (X is the cartesian product in the category of analytic spaces.) This result is used to introduce what is, in effect, a partial ordering of complex analytic singularities in terms of their complexity. This, in turn, is used to study the ways in which a germ of an analytic space may be written as the cartesian product of other germs of analytic spaces. Let F be a germ of an analytic space ( V not the reduced point). By a decomposition of V of length Received by the editors April 1, 1977. AMS (MOS) subject classifications (1970). Primary 32C15, 32C40; Secondary 32B10, 32C25.