Isosingular loci and the Cartesian product structure of complex analytic singularities

Abstract

Abstract. 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

Analytic curves in power series rings

“…We conclude by some remarks: In the absolute case S = 0 we have recovered a result of Ephraim [E,Thm. 0.2].…”

Arquile Varieties – Varieties Consisting of Power Series in a Single Variable

“…It seems that the triviality techniques from local complex analytic geometry as developed by Ephraim [Eph], extended suitably to the infinite-dimensional context, could be very appropriate to prove such type of results.…”

“…This system of equations can be formally derived with respect to and produces as in the finite-dimensional case a regular derivation of which sends all to ; cf. [Eph]. So the existence of such a derivation is a necessary criterion for the analytic triviality (it should also be sufficient, but this is not used here).…”

Untitled

“…In order to proceed inductively on the dimension of the ambient variety when working with such f 's, we quote the following direct consequence of [9], Lemmas 1.3, 1.5 (see also [7], prop. 2.4).…”