Submission date: 9 September 2010.

Abstract:

Let V be a quasi-projective algebraic variety over a non-archimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue \hat V of the Berkovich analytification V^{an} of V, and deduce several new results on Berkovich spaces from it. In particular we show that V^{an} retracts to a finite simplicial complex and is locally contractible, without any smoothness assumption on V. When V varies in an algebraic family, we show that the homotopy type of V^{an} takes only a finite number of values. The space \hat V is obtained by defining a topology on the pro-definable set of stably dominated types on V. The key result is the construction of a pro-definable strong retraction of \hat V to an o-minimal subspace, the skeleton, definably homeomorphic to a space definable over the value group with its piecewise linear structure.

Mathematics Subject Classification:

Keywords and phrases: 03C65, 03C98, 14G22 (Primary), 03C64, 14T05 (Secondary).

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