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harenbrh

Mittagsseminar zur Arithmetik: Martin Hils: Model theory of valued fields and topological tameness in non-archimedean geometry

Dienstag, 21.05.2019 10:00 im Raum SR1C
Mathematik und Informatik

In recent celebrated work, Hrushovski and Loeser developped a new model-theoretic approach to what one might call algebraic non-archimedean geometry. They show that if $V$ is a (quasi-projective) algebraic variety over a field $K$ with a complete non-archimedean norm, then its Berkovich analytification $V^{an}$ admits a strong deformation retraction onto a subspace which is homeomorphic to a finite simplicial complex. This yields various topological tameness results for $V^an$, e.g., local contractibility. The main work is done on a model-theoretic avatar $\hat{V}$ of $V^{an}$, which is a pro-definable space shown to admit a pro-definable strict deformation retraction onto a piecewise linear space, a map which then descends to the Berkovich world. In joint work with Ehud Hrushovski and Pierre Simon, we obtained a corresponding equivariant result in case $V$ is a semi-abelian variety, i.e. a pro-definmable equivariant strong deformation retraction onto a piecewise linear group. In the talk, I will give an overview of the model theory of algebraically closed valued fields which is used and explain the construction of the pro-definable space $\hat{V}$. I will then give a sketch of the proof of the main result of Hrushovski-Loeser in the case of a curve and end with a discussion of the equivariant situation in the case of a semi-abelian variety.



Angelegt am Montag, 20.05.2019 08:58 von harenbrh
Geändert am Montag, 20.05.2019 08:58 von harenbrh
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