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Books > Science & Mathematics > Mathematics > Geometry > Algebraic geometry
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Non-Archimedean Tame Topology and Stably Dominated Types (AM-192) (Hardcover)
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Non-Archimedean Tame Topology and Stably Dominated Types (AM-192) (Hardcover)
Series: Annals of Mathematics Studies
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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.
General
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