A closedness theorem over Henselian fields with analytic structure and its applications

Abstract

In this brief note, we present our closedness theorem in geometry over Henselian valued fields with analytic structure. It enables, among others, application of resolution of singularities and of transformation to normal crossings by blowing up in much the same way as over locally compact ground fields. Also given are many applications which, at the same time, provide useful tools in geometry and topology of definable sets and functions. They include several versions of the Łojasiewicz inequality, Hölder continuity of definable functions continuous on closed bounded subsets of the affine space, piecewise continuity of definable functions or curve selection. We also present our most recent research concerning definable retractions and the extension of continuous definable functions. These results were established in several successive papers of ours, and their proofs made, in particular, use of the following fundamental tools: elimination of valued field quantifiers, term structure of definable functions and b-minimal cell decomposition, due to Cluckers-Lipshitz-Robinson, relative quantifier elimination for ordered abelian groups, due to Cluckers-Halupczok, the closedness theorem as well as canonical resolution of singularities and transformation to normal crossings by blowing up due to Bierstone-Milman. As for the last tool, our approach requires its definable version established in our most recent paper within a category of definable, strong analytic manifolds and maps.