I am looking for a reference or short explanation of a proof by E. Brieskorn.

In his famous work "Singularities of complex hypersurfaces" Milnor proves that the (nowadays called) Milnor Number (in the sense of the local degree) of a polynomial $f\in\mathbb{C}{[X_1,X_2,...,X_{n+1}]}$ with an isolated singular point at the origin coincides with the $n$-th Betti number of the fibers of the corresponding Milnor fibration.

Himself remarks that Brieskorn "[...] has recently given a simple proof[...]" of this Theorem "[...] which is quite different from the one [...]" he presents and I'm looking exactly for this Proof. Milnor might mean Brieskorn's dissertation as I was told, but I do not have any access to this work. So I wonder if someone else published an at least related version of this proof.

Is it about perturbing the singularity to non-degenerate singularities?


I guess that in his book (1968) Milnor refers to the proof later published in Brieskorn's paper

"Die Monodromie der Isolierten Singularitäten von Hyperflächen", Manuscripta Mathematica 2, 103-161 (1970).

See in particular Satz 1. In fact, at the beginning of the Appendix Brieskorn writes

"Wir haben in Satz 1 bewiesen, dass der Rang der relativen de Ramschen Cohomologie einer isolierten Singularität von $f$ in $x$ gleich $b_{f, x}$ ist."

This paper is freely available on EMANI.ORG

  • $\begingroup$ Thank you, this is exactly what i was looking for. I already wondered why Milnor ignored/did'nt use holomorphic concepts... $\endgroup$ – Ben Jun 23 '11 at 12:17

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