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

Hrushovski, Ehud; Loeser, François

Hardcover

ISBN 10: 0691161682 ISBN 13: 9780691161686

Publisher: Princeton University Press, 2016

Synopsis

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.

About this title

Review:

"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

About the Author:

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. Francois Loeser is professor of mathematics at Pierre-and-Marie-Curie University in Paris.

About this edition

• Publisher: Princeton University Press
• Publication date: 2016
• ISBN 10: 0691161682
• ISBN 13: 9780691161686
• Binding: Hardcover
• Number of pages: 232