Some results of algebraic geometry over Henselian rank one valued fields

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• 作者：Krzysztof Jan Nowak
• 关键词：Closedness theorem ; Descent property for blowups ; Curve selection ; Łojasiewicz inequality ; Hereditarily rational functions ; Regulous functions and sets ; Nullstellensatz ; Cartan’s theorems A and B
• 刊名：Selecta Mathematica
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：23
• 期：1
• 页码：455-495
• 全文大小：672KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1420-9020
• 卷排序：23

文摘

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 \(K^{n} \times {\mathbb {P}}^{m}(K) \rightarrow K^{n}\) and blowups of the K-rational points of smooth K-varieties are definably closed maps; a descent property for blowups; 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.