Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

Ehud Hrushovski; François Loeser

Published by Princeton University Press 2016

Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

Ehud Hrushovski and François Loeser

Volume 192 in the series Annals of Mathematics Studies

https://doi.org/10.1515/9781400881222

Overview
Contents

About this book

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.

Author / Editor information

Ehud Hrushovski is professor of mathematics at the Hebrew University of Jerusalem. He is the coauthor of Finite Structures with Few Types (Princeton) and Stable Domination and Independence in Algebraically Closed Valued Fields. François Loeser is professor of mathematics at Pierre-and-Marie-Curie University in Paris.

Reviews

A major achievement, both in rigid algebraic geometry, and as an application of model-theoretic and stability-theoretic methods to algebraic geometry.---Anand Pillay, MathSciNet

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Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	v
1. Introduction Requires Authentication Unlicensed Licensed Download PDF	1
2. Preliminaries Requires Authentication Unlicensed Licensed Download PDF	8
3. The space v̂ of stably dominated types Requires Authentication Unlicensed Licensed Download PDF	37
4. Definable compactness Requires Authentication Unlicensed Licensed Download PDF	57
5. A closer look at the stable completion Requires Authentication Unlicensed Licensed Download PDF	70
6. Γ-internal spaces Requires Authentication Unlicensed Licensed Download PDF	76
7. Curves Requires Authentication Unlicensed Licensed Download PDF	92
8. Strongly stably dominated points Requires Authentication Unlicensed Licensed Download PDF	104
9. Specializations and ACV²F Requires Authentication Unlicensed Licensed Download PDF	119
10. Continuity of homotopies Requires Authentication Unlicensed Licensed Download PDF	142
11. The main theorem Requires Authentication Unlicensed Licensed Download PDF	154
12. The smooth case Requires Authentication Unlicensed Licensed Download PDF	177
13. An equivalence of categories Requires Authentication Unlicensed Licensed Download PDF	183
14. Applications to the topology of Berkovich spaces Requires Authentication Unlicensed Licensed Download PDF	187
Bibliography Requires Authentication Unlicensed Licensed Download PDF	207
Index Requires Authentication Unlicensed Licensed Download PDF	211
List of notations Requires Authentication Unlicensed Licensed Download PDF	215