Singularity theory

Remarks

The same definitions apply to real smooth germs, with diffeomorphisms in place of biholomorphic maps. Example: $f = x^2 + y^2$ has a singular point at $0$, since both partial derivatives vanish there. It is right equivalent to $g = xy$ via $\varphi(x, y) = (x + iy, x - iy)$, since $(x + iy)(x - iy) = x^2 + y^2$. Non-example: $f = x + y^2$ has $\partial f/\partial x = 1$ at $0$, so $0$ is not a singular point. By the holomorphic Morse lemma, every germ with a nondegenerate critical point at $0$ is right equivalent to $x_1^2 + \cdots + x_n^2$. An isolated singularity has a finite Milnor number $\mu(f) = \dim_\C \C\{x_1, \ldots, x_n\} / (\partial f/\partial x_1, \ldots, \partial f/\partial x_n)$, which is invariant under right equivalence.