---
tag: GX68S0ZH35
kind: definition
name: Singularity theory
msc: [32S05, 58K05]
defines: 42V5XGZS4F
uses: [VZXF7RNFKS, 0NAHXR1FRH, 0YTCMFB4HC]
sources:
  - url: https://en.wikipedia.org/wiki/Singularity_theory
---

## Definition

Let $f \colon (\C^n, 0) \to (\C, 0)$ be a [[germ|VZXF7RNFKS]] of a [[holomorphic function|0NAHXR1FRH]] with $f(0) = 0$.
The point $0$ is a *singular point* of $f$ if it is a [[critical point|0YTCMFB4HC]], that is, if $\partial f/\partial x_i (0) = 0$ for $i = 1, \ldots, n$.
Two such germs $f$ and $g$ are *right equivalent* if $f = g \circ \varphi$ for a germ $\varphi \colon (\C^n, 0) \to (\C^n, 0)$ of a biholomorphic map.
*Singularity theory* studies singular germs of functions, maps and varieties up to such equivalences, and their invariants.

## 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.