Several complex variables and analytic spaces Theorem

Arnold's classification of simple singularities

Builds on Isolated hypersurface singularity Ring of convergent power series Milnor number

Statement

Let $n \ge 1$ and let $\mathcal{O}_{n+1} = \C\{x, y, z_2, \dots, z_n\}$ be the ring of convergent power series in $n + 1$ variables. Two germs $f, g \in \mathcal{O}_{n+1}$ are right equivalent if $g = f \circ \phi$ for a biholomorphic germ $\phi$ of $(\C^{n+1}, 0)$ . Let $f$ be a germ with $f(0) = 0$ and an isolated singularity at $0$ . It is simple if, for $k$ large, a neighbourhood of its $k$ -jet in the space of $k$ -jets meets only finitely many right equivalence classes. Write $Q = z_2^2 + \cdots + z_n^2$ . Then $f$ is simple if and only if it is right equivalent to exactly one germ of the following list:

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

Remarks

Vladimir Arnold proved the classification in 1972. In each case the index is the Milnor number: $\mu(A_k) = \mu(D_k) = k$ and $\mu(E_k) = k$ . The proof uses finite determinacy to reduce to polynomials and the splitting lemma to remove the nondegenerate quadratic part $Q$ . Then a case analysis of the low-degree terms of the remaining function of one or two variables finishes it. A function of two variables whose $3$ -jet is zero is never simple: it lies near the family $x^4 + t x^2 y^2 + y^4$ , whose cross-ratio of the four lines is a continuous modulus.

Example: $f = x^3 + y^3$ has $\mu = 4$ and is right equivalent to $D_4$ , since its cubic part has three distinct linear factors, as does $x^2 y + y^3 = y(x^2 + y^2)$ .

Going deeper

For a student

The simple singularities appear in many guises. For $n = 2$ , they are the ADE singularities, the surface singularities $\C^2 / G$ for finite subgroups $G$ of $\mathrm{SL}_2(\C)$ , also called Kleinian or Du Val singularities. For $n = 2$ , the intersection form on $H_2$ of the Milnor fibre is, up to sign, the Cartan matrix of the Dynkin diagram of the same name. Arnold continued with the unimodal and bimodal singularities, which include the $14$ exceptional unimodal singularities. The elementary catastrophes of catastrophe theory are real forms of simple singularities of small Milnor number.

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Isolated hypersurface singularity concept
Milnor number concept
Ring of convergent power series concept
Arnold's classification of simple singularities statement

Tag	`YE5F548WMS`
URI	`m2://YE5F548WMS`
Kind	statement, theorem
Writing	W0 stub
MSC	32S05, 58K40, 14B05
Status	This copy of the site has no build data. Run `m2 status YE5F548WMS`.
About	Isolated hypersurface singularity, Singularity theory
Uses	Isolated hypersurface singularity, Ring of convergent power series, Milnor number