Non-Archimedean Tame Topology and Stably Dominated Types (AM-192) (Annals of Mathematics Studies)

Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the __p__-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed. Model-theoretic prerequisites are reviewed in the first sections. Contents 1 Introduction 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 Г-internal sets 2.9 Orthogonality to Г 2.10 V̂ for stable definable V 2.11 Decomposition of definable types 2.12 Pseudo-Galois coverings 3 The space V̂ of stably dominated types 3.1 V̂ as a pro-definable set 3.2 Some examples 3.3 The notion of a definable topological space 3.4 V̂ 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 4 Definable compactness 4.1 Definition of definable compactness 4.2 Characterization of definable compactness 5 A closer look at the stable completion 5.1 A^n and spaces of semi-lattices 5.2 A representation of P^n 5.3 Relative compactness 6 Г-internal spaces 6.1 Preliminary remarks 6.2 Topological structure of Г-internal subsets 6.3 Guessing definable maps by regular algebraic maps 6.4 Relatively Г-internal subsets 7 Curves 7.1 Definability of Ĉ 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 8 Strongly stably dominated points 8.1 Strongly stably dominated points 8.2 A Bertini theorem 8.3 Г-internal sets and strongly stably dominated points 8.4 Topological properties of V^# 9 Specializations and ACV^2F 9.1 g-topology and specialization 9.2 v-topology and specialization 9.3 ACV^2F 9.4 The map R^20 21 : V̂20 → V̂21 9.5 Relative versions 9.6 g-continuity criterion 9.7 Some applications of the continuity criteria 9.8 The v-criterion on V̂ 9.9 Definability of v- and g-criteria 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 11 The main theorem 11.1 Statement 11.2 Proof of Theorem 11.1.1: Preparation 11.3 Construction of a relative curve homotopy 11.4 The base homotopy 11.5 The tropical homotopy 11.6 End of the proof 11.7 Variation in families 12 The smooth case 12.1 Statement 12.2 Proof and remarks 13 An equivalence of categories 13.1 Statement of the equivalence of categories 13.2 Proof of the equivalence of categories 13.3 Remarks on homotopies over imaginary base sets 14 Applications to the topology of Berkovich spaces 14.1 Berkovich spaces 14.2 Retractions to skeleta 14.3 Finitely many homotopy types 14.4 More tame topological properties 14.5 The lattice completion 14.6 Berkovich points as Galois orbits Bibliography Index List of notations Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p -adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed. Model-theoretic prerequisites are reviewed in the first sections.