Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure

Krzysztof Jan Nowak

doi:10.3390/sym11070934

Abstract

We are concerned with rigid analytic geometry in the general setting of Henselian fields K with separated analytic structure, whose theory was developed by Cluckers–Lipshitz–Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore, the usual global resolution of singularities from rigid analytic geometry is no longer at our disposal. Our main purpose is to give a definable version of the canonical desingularization algorithm (the hypersurface case) due to Bierstone–Milman so that both of these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. It will be carried out within a category of definable, strong analytic manifolds and maps, which is more flexible than that of affinoid varieties and maps. Strong analytic objects are those definable ones that remain analytic over all fields elementarily equivalent to K. This condition may be regarded as a kind of symmetry imposed on ordinary analytic objects. The strong analytic category makes it possible to apply a model-theoretic compactness argument in the absence of the ordinary topological compactness. On the other hand, our closedness theorem enables application of resolution of singularities to topological problems involving the topology induced by valuation. Eventually, these three results will be applied to such issues as the existence of definable retractions or extending continuous definable functions. The established results remain valid for strictly convergent analytic structures, whose classical examples are complete, rank one valued fields with the Tate algebras of strictly convergent power series. The earlier techniques and approaches to the purely topological versions of those issues cannot be carried over to the definable settings because, among others, non-Archimedean geometry over non-locally compact fields suffers from lack of definable Skolem functions.

Highlights

We are concerned with rigid analytic geometry in the general setting of Henselian fields
K with separated analytic structure, whose theory was developed by Cluckers–Lipshitz–Robinson [1,2,3]
The main purpose of this paper is to give a definable version of the canonical desingularization algorithm due to Bierstone–Milman [14] in the hypersurface case

Summary

Introduction

We are concerned with rigid analytic geometry in the general setting of Henselian fields. Making use of the closedness theorem, it is possible to partition each ambient manifold, achieved by blowing up, into a finite number of definable clopen pieces so that, on each of them, both the exceptional hypersufaces (which reflect the history of the process and enable the further construction of the desingularization invariant) and the successive blowup, can be described in a definable geometric way This geometric bypass compensates for inability to globally describe the centers of the successive blowups in a purely analytic way, which is caused by lack of good algebraic properties of the rings of global analytic functions.

Strong Analyticity

Definable Desingularization Algorithm

Application to the Problem of Definable Retractions

Intricacies of Non-Archimedean Analytic Geometry