Global analysis, analysis on manifolds Singularities of smooth maps Theorem

Arnold's classification of simple critical points

Builds on Germ Critical point Holomorphic function Modality of a singularity

Statement

Let $f \colon (\C^n, 0) \to (\C, 0)$ be a germ of a holomorphic function with a critical point at $0$ . Call $f$ simple if, in a jet space of high enough order, some neighbourhood of $f$ meets only finitely many right equivalence classes. Equivalently, $f$ has modality $0$ . Then $f$ is simple if and only if it has the following form in some holomorphic coordinates $(x, y, z_1, \ldots, z_{n-2})$ centred at $0$ . It is one of the following germs plus $z_1^2 + \cdots + z_{n-2}^2$ , with the obvious changes when $n \le 2$ :

$A_k$ , $k \ge 1$ : $x^{k+1} + y^2$ ;
$D_k$ , $k \ge 4$ : $x^2 y + y^{k-1}$ ;
$E_6$ : $x^3 + y^4$ ;
$E_7$ : $x^3 + x y^3$ ;
$E_8$ : $x^3 + y^5$ .

Remarks

For $n = 1$ only the germs $A_k$ , written $x^{k+1}$ , occur. In each case the index is the Milnor number, $\dim_\C \mathcal{O}_n / J(f)$ , where $\mathcal{O}_n$ is the ring of convergent power series and $J(f)$ is the ideal of partial derivatives. The proof proceeds by the corank and the lowest-order terms of $f$ , using finite determinacy to reduce to polynomials. It shows that every germ outside the list deforms, by arbitrarily small perturbations, into members of a unimodal family. These families are $x^3 + y^3 + z^3 + a xyz$ , $x^4 + y^4 + a x^2 y^2$ and $x^3 + y^6 + a x^2 y^2$ , with suitable conditions on $a$ . Example: $x^2 y + y^3$ is a $D_4$ singularity; its cubic part factors into three distinct lines through $0$ . Over $\R$ the same list holds up to signs of the terms. For $n = 3$ the zero sets of these germs are the ADE singularities of surfaces.

Going deeper

For a student

The theorem is the start of Arnold's classification of critical points by modality, which continues with the unimodal and bimodal families. The ADE pattern links it with finite subgroups of $SL(2, \C)$ , simple Lie algebras and the Du Val surface singularities. The real forms of these germs with small codimension are the organising centres of the elementary catastrophes in Thom's classification.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Critical point concept
Germ concept
Holomorphic function concept
Modality of a singularity concept
Arnold's classification of simple critical points statement

Tag	`M5437FP73G`
URI	`m2://M5437FP73G`
Kind	statement, theorem
Writing	W0 stub
MSC	58K40, 32S05, 14B05
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About	Germ, Critical point
Uses	Germ, Critical point, Holomorphic function, Modality of a singularity