On globally defined semianalytic sets

Abstract

In this work we present the concept of C-semianalytic subset of a real analytic manifold and more generally of a real analytic space. C-semianalytic sets can be understood as the natural generalization to the semianalytic setting of global analytic sets introduced by Cartan (C-analytic sets for short). More precisely S is a C-semianalytic subset of a real analytic space $(X,{\mathcal {O}}_X)$ if each point of X has a neighborhood U such that $S\cap U$ is a finite boolean combinations of global analytic equalities and strict inequalities on X. By means of paracompactness C-semianalytic sets are the locally finite unions of finite boolean combinations of global analytic equalities and strict inequalities on X. The family of C-semianalytic sets is closed under the same operations as the family of semianalytic sets: locally finite unions and intersections, complement, closure, interior, connected components, inverse images under analytic maps, sets of points of dimension k, etc. although they are defined involving only global analytic functions. In addition, we characterize subanalytic sets as the images under proper analytic maps of C-semianalytic sets. We prove also that the image of a C-semianalytic set S under a proper holomorphic map between Stein spaces is again a C-semianalytic set. The previous result allows us to understand better the structure of the set N(X) of points of non-coherence of a C-analytic subset X of a real analytic manifold M. We provide a global geometric-topological description of N(X) inspired by the corresponding local one for analytic sets due to Tancredi and Tognoli (Riv Mat Univ Parma (4) 6:401–405, 1980), which requires complex analytic normalization. As a consequence it holds that N(X) is a C-semianalytic set of dimension ${\le }\dim (X)-2$.