Videos

A Model Theoretic Approach to Berkovich Spaces

Presenter
February 4, 2014
Keywords:
  • ACVF
  • Berkovich spaces
  • definable types
  • orthogonality
MSC:
  • 12J10
  • 12Jxx
  • 37P50
  • 37P55
  • 37Pxx
  • 03Cxx
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).
Supplementary Materials