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Preprint Number 1229

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1229. Krzysztof Jan Nowak
The closedness theorem and its applications in algebraic geometry over Henselian valued fields
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Submission date: 4 June 2017

Abstract:

We develop geometry of algebraic subvarieties of K^{n} over arbitrary Henselian valued fields K of equicharacteristic zero. This is a continuation of our previous article, devoted to algebraic geometry over rank one valued fields, which in general requires more involved techniques and to some extent new treatment. Again, at the center of our approach is the closedness theorem that the projections K^{n} × P^{m}(K) → K^{n} are definably closed maps. Hence we obtain, in particular, a descent property for blow-ups, which enables application of resolution of singularities in much the same way as over the locally compact ground field. As before, the proof of that theorem uses i.a. the local behaviour of definable functions of one variable and fiber shrinking, a relaxed version of curve selection. But now, to achieve the former result, we first examine functions given by algebraic power series. The results from our previous article will be established in the general settings: several versions of curve selection (via resolution of singularities) and of the Łojasiewicz inequality (via two instances of quantifier elimination indicated below), extending continuous hereditarily rational functions and the theory of regulous functions, sets and sheaves, including Nullstellensatz and Cartan's theorems A and B. Two basic tools applied in this paper are quantifier elimination for Henselian valued fields due to Pas and relative quantifier elimination for ordered abelian groups (in a many-sorted language with imaginary auxiliary sorts) due to Cluckers--Halupczok. Other applications of the closedness theorem are piecewise continuity of definable functions and Hölder continuity of definable functions on closed bounded subsets of K^{n}.

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Full text arXiv 1706.01774: pdf, ps.


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