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The answer is in fact no.

A complex variety $X$ can never be a differentiable manifold (not even of class $C^1$) throughout a neighborhood of a singular point.

You can find a proof in Milnor's book "Singular Points of Complex Hypersurfaces"Hypersurfaces", Annals of Mathematics Studies 61, remark at page 13.

Notice that $X$ can be a topological manifold (i.e., a manifold of class $C^0$) in a neighborhood of a singular point. For instance, the cuspidal plane cubic $y^2z=x^3$ is homeomorphic to $\mathbb{P}^1$.

3 added 218 characters in body

The answer is in fact no.

A complex variety $X$ can never be a smooth differentiable manifold (not even of class $C^1$) throughout a neighborhood of a singular point.

You can find a proof in Milnor's book "Singular Points of Complex Hypersurfaces", Annals of Mathematics Studies 61, remark at page 13.

Notice that $X$ can be a topological manifold (i.e., a manifold of class $C^0$) in a neighborhood of a singular point. For instance, the cuspidal plane cubic $y^2z=x^3$ is homeomorphic to $\mathbb{P}^1$.

2 added 47 characters in body

The answer is in fact no.

A complex variety can never be a smooth manifold (not even of class $C^1$) throughout a neighborhood of a singular point.

You can find a proof in [Milnor, Milnor's book "Singular Points of Complex HypersurfacesHypersurfaces", Remark Page 13]Annals of Mathematics Studies 61, remark at page 13.

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