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A Model Theoretic Approach to Berkovich Spaces

Introductory Workshop: Model Theory, Arithmetic Geometry and Number Theory February 03, 2014 - February 07, 2014

February 04, 2014 (02:30 PM PST - 03:30 PM PST)
Speaker(s): Martin Hils (Westfälische Wilhelms-Universität Münster)
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC



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).

20078?type=thumb Hils notes 1.51 MB application/pdf Download
Video/Audio Files


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