A Model Theoretic Approach to Berkovich Spaces
Introductory Workshop: Model Theory, Arithmetic Geometry and Number Theory February 03, 2014  February 07, 2014
Location: SLMath: Eisenbud Auditorium
v1249
If K is a (complete) eld with respect to a nonarchimedean 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 complexanalytic case. One approach to overcome this problem is due to Berkovich in the late 80's. He develops a theory of Kanalytic spaces, adding points to the set of naive points. In particular, for an algebraic variety V dened over K he constructs what is now called its Berkovich analytication 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)denable space bV which is a close analogue of V an, and they establish strong topological tame ness properties in this denable setting, combining ominimality 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 analytication of an algebraic variety. I will then explain how the model theory of ACVF, in particular the theory of stable domination (due to HaskellHrushovskiMacpherson 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 denability, I will sketch HrushovskiLoeser's construction of a strong de formation retraction of bV onto a piecewise linear subspace (in the denable category).
Hils notes

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