The invariance of Milnor’s number implies the invariance of the topological type. (English) Zbl 0351.32009

Let \(F(t,z)\) be a polynomial in \(z=(z_0,z_1,\dots,z_n)\) with coefficients which are smooth complex valued functions of \(t\in I= [0,1]\) such that \(F(t,0)=0\) and that for each \(t\in I\), the polynomials \((\partial F/\partial z_i)(t,z)\) in \(z\) have an isolated zero at \(0\). Assume more over that the integer \(\mu_t=\dim_{\mathbb{C}}\mathbb{C}/(\partial f/\partial z_0)(t,z),\dots,(\partial f/\partial z_n)(t,z))\) is independent of \(t\). The authors prove that the monodromy fibrations of the singularities of \(F(0,z) =0\) and \(F(1, z) =0\) at \(0\) are of the same fiber homotopy and, if further \(n\neq 2\), these fibrations are even differentially isomorphic and the topological types of the singularities are the same. The hypothesis \(n\neq 2\) comes from using h-cobordism theorem. This gives a proof of the Hironaka’s conjecture for \(n=1\) in the more general case of a \(C^{\infty}\) family of \(n\)-dimensional hypersurfaces of dimension \(n\neq 2\). Making use of the results of K. Brauner [Abh. math. Semin. Hamburg Univ. 6, 1–55 (1928; JFM 54.0373.01)], W. Burau [Abh. Math. Semin. Hamb. Univ. 9, 125–133 (1932; Zbl 0006.03402; JFM 58.0615.01)] and O. Zariski [Am. J. Math. 54, 453–465 (1932; Zbl 0004.36902; JFM 58.0614.02)], the authors prove that Puiseux pairs of an analytically irreducible plane curve singularity depends only on the topology of the singularity.


32Sxx Complex singularities
14B05 Singularities in algebraic geometry
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