I am looking for examples of normal complex spaces $X$ which locally around a singular point are homeomorphic to a smooth complex manifold.
The only example I know is a curve with a cusp, but this is of course not normal.
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6$\begingroup$ See Brieskorn, "Examples of singular normal complex spaces which are topological manifolds", Proc Nat Acad Sci. $\endgroup$– Donu ArapuraCommented Oct 23, 2020 at 13:33
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3$\begingroup$ In a similar vein as Donu's comment, check also Milnor's "Singular points of complex hypersurfaces", in particular Thm 2.10 and then chapter 8. Also remark that isolated singular points of hypersurfaces of dim $\ge 2$ are normal. $\endgroup$– ChrisCommented Oct 23, 2020 at 13:52
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4$\begingroup$ Actually Brieskorn's examples are so simple that it is worth to recall them: he takes $X\subset \mathbb{C}^{2n}$ defined by $z_1^{3}+z_2^2.\ldots +z_{2n}^2=0$. $\endgroup$– abxCommented Oct 23, 2020 at 13:54
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$\begingroup$ Are these the only examples?? Nothing else?? quite surprising! In case, you could turn the comments into an answer. $\endgroup$– GiulioCommented Nov 5, 2020 at 15:17
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