Abstract In this note, we study the relationship between Zariski and relative closedness for actions of (smooth) algebraic groups defined over valued (mainly local) fields of any characteristic. In particular, we use some recent basic results regarding the completely reducible subgroups and cocharacter-closedness due to Bate–Herpel–Röhrle–Tange and Uchiyama to construct some actions of simple algebraic groups G of the types D 4 {D_{4}} , E 6 {E_{6}} , E 7 {E_{7}} , E 8 {E_{8}} , G 2 {G_{2}} on an affine variety defined over a local function field k, and v ∈ V ⁢ ( k ) {v\in V(k)} such that the geometric orbit G . v {G.v} is Zariski closed although the corresponding relative orbit G ⁢ ( k ) . v {G(k).v} is not closed in the topology induced from k. Besides, by using an interesting result due to Gabber, Gille and Moret-Bailly, we show that this phenomenon does not appear when we consider the action of either a smooth unipotent group or a smooth commutative algebraic group, defined over an admissible valued (e.g., local) field.

中文翻译：

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关于值域上代数群作用的几何轨道和有理轨道的拓扑结构

摘要在这篇笔记中，我们研究了 Zariski 与在任何特征的值（主要是局部）域上定义的（平滑）代数群的动作的相对封闭性之间的关系。特别是，我们使用最近关于完全可约子群和由于 Bate-Herpel-Röhrle-Tange 和 Uchiyama 的共字符封闭性的一些基本结果来构造 D 4 {D_{4}} 类型的简单代数群 G 的一些动作， E 6 {E_{6}} , E 7 {E_{7}} , E 8 {E_{8}} , G 2 {G_{2}} 在局部函数域 k 上定义的仿射变体上，并且 v ∈ V ⁢ ( k ) {v\in V(k)} 使得几何轨道 G . v {Gv} 是 Zariski 闭的，尽管相应的相对轨道 G ⁢ ( k ) 。v {G(k).v} 在由 k 导出的拓扑中不是封闭的。此外，通过使用 Gabber、Gille 和 Moret-Bailly 的有趣结果，