A Model Theoretic Approach to Berkovich Spaces

If K is a (complete) eld with respect to a non-archimedean ab- solute value, then K is totally disconnected as a topological eld. This is a serious obstacle when one wants to develop analytic geometry over K as this is done in the complex-analytic case. One approach to overcome this problem is due to Berkovich in the late 80's. He develops a theory of K-analytic spaces, adding points to the set of naive points. In particular, for an algebraic variety V dened over K he constructs what is now called its Berkovich analytication V an which contains V (K) (with its natural topology) as a dense subspace and which is locally compact and locally arcwise connected. Recently, using the geometric model theory of algebraically closed valued elds (ACVF), Hrushovski and Loeser constructed a (pro-)denable space bV which is a close analogue of V an, and they establish strong topological tame- ness properties in this denable setting, combining o-minimality with tools from stability theory. These properties are then shown to transfer to the ac- tual Berkovich setting. In the tutorial, I will rst introduce the notion of a Berkovich space, em- phasising the analytication of an algebraic variety. I will then explain how the model theory of ACVF, in particular the theory of stable domination (due to Haskell-Hrushovski-Macpherson and which will be discussed in Deirdre Haskell's talk), is used in the work of Hrushovski and Loeser to construct the space bV . Finally, in the case where V is an algebraic curve, stressing the role of denability, I will sketch Hrushovski-Loeser's construction of a strong de- formation retraction of bV onto a piecewise linear subspace (in the denable category).

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