Some results of algebraic geometry over Henselian rank one valued fields

Abstract

We develop geometry of affine algebraic varieties in $K^n$ over Henselian rank one valued fields $K$ of equicharacteristic zero. Several results are provided including: the projection $K^n\times {\mathbb{P}}^m(K)\to K^n$ and blow-ups of the $K$-rational points of smooth $K$-varieties are definably closed maps, a descent property for blow-ups, curve selection for definable sets, a general version of the Łojasiewicz inequality for continuous definable functions on subsets locally closed in the $K$-topology and extending continuous hereditarily rational functions, established for the real and $p$-adic varieties in our joint paper with J. Kollár. The descent property enables application of resolution of singularities and transformation to a normal crossing by blowing up in much the same way as over the locally compact ground field. Our approach relies on quantifier elimination due to Pas and a concept of fiber shrinking for definable sets, which is a relaxed version of curve selection. The last three sections are devoted to the theory of regulous functions and sets over such valued fields. Regulous geometry over the real ground field $\mathbb{R}$ was developed by Fichou--Huisman--Mangolte--Monnier. The main results here are regulous versions of Nullstellensatz and Cartan's Theorems A and B.

This paper has been published in Selecta Mathematica

References in corpus (3)

• Stratified-algebraic vector bundles
• Some results from algebraic geometry over complete discretely valued fields
• Some results from algebraic geometry over Henselian real valued fields

Cited by in corpus (8)

• Curve-rational functions
• Motivic local density
• Definable retractions and a non-Archimedean Tietze--Urysohn theorem over Henselian valued fields
• Definable transformation to normal crossings over Henselian fields with separated analytic structure
• Definable retractions over complete fields with separated power series
• On functions given by algebraic power series over Henselian valued fields
• Blown-up Čech cohomology and Cartan's Theorem B on real algebraic varieties
• Tame topology in Hensel minimal structures