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

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.

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

“…Fix a Henselian non-trivially valued field K of equicharacteristic zero, with analytic structure (cf. [7,8,24,26]) and the analytic language L being an analytic expansion of the 3-sorted one (Denef-Pas [29]) or the 2-sorted one (Basarab-Kuhlmann [2,19]).…”

“…Finally, in the last section, we state some results concerning definable retractions and the extension of continuous definable functions. They were established in papers [24,27,28]. Their proofs rely basically on our closedness theorem and on canonical resolution of singularities and transformation to normal crossings by blowing up due to Bierstone-Milman [3].…”

“…Their proofs rely basically on our closedness theorem and on canonical resolution of singularities and transformation to normal crossings by blowing up due to Bierstone-Milman [3]. As for the last tool, our approach requires its definable version established in our most recent paper [24] within a category of definable, strong analytic manifolds and maps.…”

“…In the recent papers [24,27,28], we provided some theorems on the existence of definable retractions. They immediately yield some definable, non-Archimedean versions of the classical theorems on extending continuous functions as the theorems of Tietze-Urysohn or Dugundji (cf.…”

“…To go around the above problem, in our last paper [24] we adapted to the definable settings the canonical algorithm for resolution of singularities due to Bierstone-Milman [3]. That algorithm provides a local invariant such that blowing up its maximum strata leads to desingularization or transformation to normal crossings.…”