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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 06, 2014 (09:30 AM PST - 10:30 AM PST)
Speaker(s): Martin Hils (Université de Paris VII (Denis Diderot))
Location: MSRI: Simons Auditorium
Video

Abstract

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).

Supplements
20171?type=thumb Hils Notes 1.81 MB application/pdf Download
Video/Audio Files

V1254

H.264 Video v1254.mp4 374 MB video/mp4 rtsp://videos.msri.org/data/000/019/887/original/v1254.mp4 Download
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