Flattening and subanalytic sets in rigid analytic geometry. (Q2766435)

Subanalytic sets arise naturally in real analytic geometry as images of proper analytic maps. The structure of such an image can be quite complicated: it is not necessarily definable by means of inequalities between analytic functions, that is, it is not in general semianalytic. Therefore, a compact subset of a real analytic manifold is called subanalytic if it is (at least locally) the projection of a relatively compact semianalytic set.NEWLINENEWLINEReal analytic sets were first introduced by Łojasiewicz and subsequently studied by \textit{A. M. Gabrielov} in [Funkts. Anal. Prilozh. 2, No. 4, 18--30 (1968; Zbl 0179.08503)] and \textit{H. Hironaka} in [Astérisque 7--8 (1973), 415--440 (1974; Zbl 0287.14006), Number Theory, Algebr. Geom., Commut. Algebra, in Honor of Yasuo Akizuki, 453--493 (1973; Zbl 0297.32008)] by complex analytic methods (fattening, Voûte Étoilée) and by geometric techniques (Resolution of Singularities). A new approach appeared in a paper [Ann. Math. (2) 128, No. 1, 79--138 (1988; Zbl 0693.14012)] by \textit{J. Denef} and \textit{L. van den Dries}, where a model-theoretic point of view was taken. This resulted in a much more concise formulation which has the enormous advantage of being applicable in the p-adic context. Motivated by problems of elliptic curves, Tate constructed a theory of rigid analytic geometry over complete non-archimedean algebraically closed fields. This theory was further developed by Kiehl, Grauert et al., largely in analogy with complex analysis. A little later Raynaud gave an alternative treatment through formal schemes and more recently still, Berkovich approached the subject from the point of view of spectral theory.NEWLINENEWLINEIt was the insight of Denef that the methods of Hironaka might be used in the rigid case as well. The key observation is a result due to Raynaud and Gruson describing the image of a flat map between affinoid varieties; this serves as a replacement for the Fibre Cutting Lemma of flat maps in Hironaka's work. To make a reduction to the flat case, one needs a good theory of rigid analytic flatificators (to be used as centres of local blowing-ups) and the construction of the Voûte Étoilée (a compact Hausdorff space encoding finite sequences of local blowing-ups; this is the analytic version of the Zariski-Riemann manifold). The former is carried out by the second author in [Q. J. Math., Oxf. II. Ser. 50, No. 199, 321--353 (1999; Zbl 0953.32017)] and by the first author in [Proc. Lond. Math. Soc., III. Ser. 80, No. 1, 179--197 (2000; Zbl 1029.32009)].NEWLINENEWLINEThe present paper will put all these results together to obtain the sought-for theory of rigid subanalytic sets. The main theorem states that any rigid subanalytic set can be described by inequalities among functions which are obtained by composition and division of analytic functions.