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 \L{}ojasiewicz 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. Koll\'ar. 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.