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

1. Auflage 2016
ISBN: 978-1-4008-8122-2
Verlag: De Gruyter
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, Band 192, 232 Seiten, EPUB

Reihe: 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.

Autoren/Hrsg.

Hrushovski, Ehud

Loeser, François

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Analytische Geometrie
• Mathematik | Informatik Mathematik Topologie
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie

Inhaltsverzeichnis

Frontmatter, pg. i

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

Über Autor(innen)

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.