Let \(G\) be a complex connected reductive algebraic group. A Jordan class in the corresponding Lie algebra \(\mathfrak{g}\) of the element \(x=x_s+x_n\) consists of all elements whose centralisers are \(G\)-conjugate to the centralizer \(\mathfrak{c}_ \mathfrak{g}(x)\). The Jordan classes for \(G\) are defined similarly, but more cumbersomely. Jordan classes in a reductive group or Lie algebra are locally closed, smooth, irreducible \(G\)-stable subsets of elements having similar Jordan decomposition.
It is proved here that the closure of every Jordan class \(J\) in \(G\) at a point \(x\) with Jordan decomposition \(x = rv\) (where the element \(r\) is semisimple, \(v\) is unipotent) 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. This allows to reduce the local study of Jordan classes around any element to a local study around a unipotent one. For unipotent \(x\) it is proved that the closure of \(J\) around \(x\) is smoothly equivalent to the closure of a Jordan class in \(\mathfrak g\) around \(\exp^{-1} x\). As an applications for some simple and simply connected groups \(G\) a list of smooth sheets in \(G\) and the complete list of regular Jordan classes whose closure is normal and Cohen-Macaulay are given. Then the special case \(G=\mathrm{SL}(n,\mathbf C)\) is considered.