Non-archimedean tame topology and stably dominated types

Abstract

Let V be a quasi-projective algebraic variety over a nonarchimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue V of the Berkovich analytification V an of V , and deduce several new results on Berkovich spaces from it. In particular we show that V an retracts to a finite simplicial complex and is locally contractible, without any smoothness assumption on V . When V varies in an algebraic family, we show that the homotopy type of V an takes only a finite number of values. The space V is obtained by defining a topology on the pro-definable set of stably dominated types on V . The key result is the construction of a pro-definable strong retraction of V to an o-minimal subspace, the skeleton, definably homeomorphic to a space definable over the value group with its piecewise linear structure. ContentsChapter 1. Introduction Chapter 2. Preliminaries 2.1. Definable sets 2.2. Pro-definable and ind-definable sets 2.3. Definable types 2.4. Stable embeddedness 2.5. Orthogonality to a definable set 2.6. Stable domination 2.7. Review of ACVF 2.8. Γ-internal sets 2.9. Orthogonality to Γ 2.10. " V for stable definable V 2.11. Decomposition of definable types 2.12. Pseudo-Galois coverings Chapter 3. The space "V of stably dominated types 3.1. "V as a pro-definable set 3.2. Some examples 3.3. The notion of a definable topological space 3.4. "V as a topological space 3.5. The affine case 3.6. Simple points 3.7. v-open and g-open subsets, v+g-continuity 3.8. Canonical extensions 3.9. Paths and homotopies 3.10. Good metrics 3.11. Zariski topology 3.12. Schematic distance Chapter 4. Definable compactness 4.1. Definition of definable compactness 4.2. Characterization of definable compactness Chapter 5. A closer look at the stable completion iii iv CONTENTS 5.1. " A n and spaces of semi-lattices 5.2. A representation of " P n 5.3. Relative compactness Chapter 6. Γ-internal spaces 6.1. Preliminary remarks 6.2. Topological structure of Γ-internal subsets 6.3. Guessing definable maps by regular algebraic maps 6.4. Relatively Γ-internal subsets Chapter 7. Curves 7.1. Definability of " C for a curve C 7.2. Definable types on curves 7.3. Lifting paths 7.4. Branching points 7.5. Construction of a deformation retraction Chapter 8. Strongly stably dominated points 8.1. Strongly stably dominated points 8.2. A Bertini theorem 8.3. Γ-internal sets and strongly stably dominated points 8.4. Topological properties of V # Chapter 9. Specializations and ACV 2 F 9.1. g-topology and specialization 9.2. v-topology and specialization 9.3. ACV 2 F 9.4. The map R 20 21 : " V 20 → " V 21 9.5. Relative versions 9.6. g-continuity criterion 9.7. Some applications of the continuity criteria 9.8. The v-criterion on " V 9.9. Definability of v-and g-criteria Chapter 10. Continuity of homotopies 10.1. Preliminaries 10.2. Continuity on relative P 1 10.3. The inflation homotopy 10.4. Connectedness and the Zariski topology Chapter 11. The main theorem 11.1. Statement 11.2. Proof of Theorem 11.1.1: Preparatio...

Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry

“…For instance, while it has been known for some time that the analytification of a smooth variety over a trivially normed field is contractible, the only known proof that they are locally contractible is quite recent and passes through model theory and spaces of stably dominated types. This local contractibility is just one of the fundamental consequences of the tameness theorem of Hrushovski and Loeser [HL12].…”

“…In particular, the proof of the tameness theorem for a single variety in dimension n requires a tameness statement for families of varieties in lower dimensions. See the Bourbaki notes of Ducros [Duc13] for an excellent introduction to this work, and the original paper [HL12] for further details. 4.4.…”

Definable functions continuous on curves in o-minimal structures

“…Given an n-tuple, c, and set A, there are certain points that must be in any closed A-definable set containing c, namely the i-closures defined in this section. The iclosure of c for each i ≤ n is the limit (in the sense of [HL10]) of tp(c ≥Q(i) /Ac <Q(i) ). A principle of our proof of Theorem A will be that if a function can be continuously extended to the i-closure points for all i, then it can be continuously extended to an A-definable closed set containing c. We can assure continuity on i-closures by bounding the various values a function takes by another function that goes to 0 as it approaches an i-closure point.…”

“…The way that such a set containing a type can fail to exist is that, in some sense, the type lies in a "gap" between two regions on which the function takes very different values, but which share a common boundary point that is the limit of the type, in the sense of [HL10].…”

Nonarchimedean geometry of Witt vectors

“…For instance, Berkovich showed that smooth analytic spaces over a field are locally contractible [4,5]. More recently, Payne [26] showed that the analytification of an algebraic variety over a field can be viewed as an inverse limit of finite polyhedral complexes; separately, Hrushovski and Loeser [16] have used model-theoretic techniques to show that such analytifications are locally contractible and retract onto finite CW-complexes. One can also relate homotopy types of analytic spaces to degenerations; for instance, the analytification of a semistable curve over a complete discretely valued field has the same homotopy type as the graph of the special fibre of a minimal proper regular model over the valuation subring.…”