We prove that the closure of every Jordan class $J$ in a semisimple simply connected complex algebraic group $G$ at a point $x$ with Jordan decomposition $x=rv$ is smoothly equivalent to the union of closures of those Jordan classes in the centraliser of $r$ that are contained in $J$ and contain $x$ in their closure. For $x$ unipotent, we also show that the closure of $J$ around $x$ is smoothly equivalent to the closure of a Jordan class in $\mathrm{Lie}(G)$ around ${\exp }^{-1}x$. For $G$ simple we apply these results in order to determine a (non‐exhaustive) list of smooth sheets in $G$, the complete list of regular Jordan classes whose closure is normal and Cohen–Macaulay, and to prove that all sheets and Lusztig strata in ${\mathrm{SL}}_n(\mathbb{C})$ are smooth.

中文翻译：

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半简单代数组中约旦类的局部几何

我们证明约旦每堂课的结束 $Ĵ$ 在一个半简单的简单连通的复代数群中 $G$ 在某一点上 $X$ 用约旦分解 $X=\mathrm{[R}v$ 平滑地相当于在中央集权者中关闭那些约旦阶级的联合 $\mathrm{[R}$ 包含在 $Ĵ$ 并包含 $X$在他们关闭。为了$X$ 单能的，我们还证明了封闭 $Ĵ$ 大约 $X$ 顺利地相当于关闭了乔丹班 $\mathrm{说谎}（G）$ 大约 ${\mathrm{经验值}}^{-\mathrm{1个}}X$。为了$G$ 很简单，我们应用这些结果来确定（非穷举的）光滑纸列表 $G$，常规乔丹班级的完整列表，这些班级的班级正常且科恩–麦克劳莱关闭，并证明所有表层和Lusztig地层均处于 ${\mathrm{SL}}_{ñ}（\mathbb{C}）$ 顺利。