Complex analytic space

Abstract: Inspired by Zariski's saturation, we associate to each reduced equidimensional germ (X,x) of a complex analytic space the algebra of its meromorphic functions satisfying a Lipschitz condition. We show that it is a local analytic subalgebra of the integral closure of the analytic algebra corresponding to (X,x). In some cases, for example for hypersurfaces, it coincides with Zariski's saturation. Our construction, which relies on the normalized blowing-up of the diagonal in the product XxX, also gives a coordinate-free description of the "transversal" Puiseux characteristic exponents of a plane branch.

Abstract: A criterion is given for a strong type in a finite rank stable theory $T$ to be (almost) internal to a given nonmodular minimal type. The motivation comes from results of Campana which give criteria for a compact complex analytic space to be “algebraic” (namely Moishezon). The canonical base property for a stable theory states that the type of the canonical base of a stationary type over a realisation is almost internal to the minimal types of the theory. It is conjectured that every finite rank stable theory has the canonical base property. It is shown here, that in a theory with the canonical base property, if $p$ is a stationary type for which there exists a family of types $q_b$, each internal to a nonlocally modular minimal type $r$, and such that any pair of independent realisations of $p$ are “connected” by the $q_b$’s, then $p$ is almost internal to $r$.

Abstract: Let X be a (not necessarily reduced) complex analytic space, and let V be a germ of an analytic space. The locus of points q in X at which the germ Xq is complex analytically isomorphic to V 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. For p E X let Iso(X,p) = {q E XlXq ;)X. (~ here and elsewhere will mean complex analytically isomorphic.) It will be shown that: THEOREM 0.2. For any p E X, Iso(X, p) is a (possibly 0-dimensional) complex submanifold of some open subset of X. Moreover, for any q E Iso(X, p) there is an open neighbornood U of q, and an analytic space Y such that U_ Y x (U n Iso(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 V 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 32B 10, 32C25.

Abstract: In this paper we are interested in the following problem: Suppose V is an analytic subvariety of a (not necessarily reduced) complex analytic space X, F is a coherent analytic sheaf on XV, and 0: XV -X is the inclusion map. When is Gq(Y) coherent (where Gq(Y) is the qth direct image of Y under 0)? The case q = 0 is very closely related to the problem of extending Y' to a coherent analytic sheaf on X. This problem of extension has already been dealt with in Frisch-Guenot [1], Serre [9], Siu [11]-[14], Thimm [17], and Trautmann [18]-[20]. So, in our investigation we assume that Y' admits a coherent analytic extension on all of X. In reponse to a question of Serre f9, p. 366], Tratumann has obtained the following in [21]: