On globally defined semianalytic sets

• Published in: Mathematische annalen 2016-10, Vol.366 (1-2), p.613-654
• Main Authors: Acquistapace, Francesca, Broglia, Fabrizio, Fernando, José F.
• Format: Article
• Language: English
• Subjects: Analytic functions; Boolean algebra; Inequalities; Intersections; Manifolds (mathematics); Mathematical analysis; Mathematics; Mathematics and Statistics; Set theory; Unions
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-015-1342-5

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 , O X ) if each point of X has a neighborhood U such that S ∩ 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 ≤ dim ( X ) - 2 .