Several complex variables and analytic spaces Complex singularities Theorem

Lê–Ramanujam theorem

Builds on Isolated hypersurface singularity Milnor number Homeomorphism

Statement

Let $n \ne 2$ be a non-negative integer. Let $F(z, t)$ be a holomorphic function on a neighbourhood of $(0, 0)$ in $\C^{n+1} \times \C$ , and write $f_t(z) = F(z, t)$ . Assume that, for all $t$ in a neighbourhood of $0$ , $f_t(0) = 0$ and $f_t$ has an isolated singularity at $0$ . Assume also that the Milnor number $\mu(f_t)$ does not depend on $t$ . Then, for all $t$ small enough, there is a homeomorphism $\phi$ between neighbourhoods of $0$ in $\C^{n+1}$ with $\phi(0) = 0$ and $\phi(f_0^{-1}(0)) = f_t^{-1}(0)$ . In words, a $\mu$ -constant family of isolated hypersurface singularities has constant topological type.

Remarks

Lê Dũng Tráng and C. P. Ramanujam proved the theorem in 1976. The proof shows that the Milnor fibrations of $f_0$ and $f_t$ are related by an h-cobordism, and then applies the h-cobordism theorem of Smale. That theorem needs the cobordism to have dimension at least $6$ , which is why the case $n = 2$ , surfaces in $\C^3$ , is excluded; this case is still open. The converse holds: the Milnor number is a topological invariant of the hypersurface germ, so a family of constant topological type has constant $\mu$ .

Example: $f_t(x, y) = x^4 + y^4 + t x^2 y^2$ for $t$ near $0$ . For $t \ne \pm 2$ the quartic form $f_t$ is a product of four distinct linear factors, so $f_t$ has an isolated singularity with $\mu(f_t) = 9$ . The theorem says that all these germs are topologically the same as $x^4 + y^4$ , four lines through the origin.

Going deeper

For a student

The theorem is a central result of equisingularity theory, which asks when a family of singularities should count as having the same singularity. For plane curves, $n = 1$ , constant $\mu$ is equivalent to constant topological type, and to constant Puiseux characteristics. Later work showed that a $\mu$ -constant family is in fact topologically trivial as a family when $n \ne 2$ . The Milnor number is upper semicontinuous in families, so it can only jump up at special parameter values.

Reading path

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Homeomorphism concept
Isolated hypersurface singularity concept
Milnor number concept
Lê–Ramanujam theorem statement

Tag	`JRAEY07Y4G`
URI	`m2://JRAEY07Y4G`
Kind	statement, theorem
Writing	W0 stub
MSC	32S15, 32S05, 32S50
Status	This copy of the site has no build data. Run `m2 status JRAEY07Y4G`.
About	Milnor number, Isolated hypersurface singularity
Uses	Isolated hypersurface singularity, Milnor number, Homeomorphism