Characterization of isolated homogeneous hypersurface singularities in \(\mathbb C^{4}\)
Let \((V,o)\subset (\mathbb C^{n+1},o)\) be an isolated hypersurface singularity defined by a holomorphic function \(f\). It is a natural question to ask when \(V\) is weighted homogeneous or homogeneous. The famous theorem of \textit{K. Saito} [Invent. Math. 14, 123--142 (1971; Zbl 0224.32011)] states that \(V\) is weighted homogeneous if and only if the Milnor number \(\mu\) and the Tjurina number \(\tau\) of \(V\) coincide. Let \(\nu\) and \(p_g\) denote the multiplicity and the geometric genus of \(V\), respectively. The main theorem of this article is: If \(n=3\) and \(f\) is a weighted homogeneous polynomial, then \(\mu\geq 24 p_g+ 2\nu ^3-5\nu^2+2\nu+1\), and the equality holds if and only if \(f\) is homogeneous. This result and Saito's theorem give a characterization of homogeneous singularities in terms of the above invariants. Note that these invariants are computed from the weights. The main theorem for the case \(p_g>0\) and related results have already been proved by \textit{K. Lin} and \textit{S. S.-T. Yau} [Commun. Anal. Geom. 12, No. 5, 1201--1219 (2004; Zbl 1072.32022)]. These results reduce the question to proving an inequality \(\mu>16\) under suitable conditions. The proof is based on the classification of weighted homogeneous singularities [\textit{S. S.-T. Yau} and \textit{Y. Yu}, Math. Nachr. 246--247, 207--233 (2002; Zbl 1019.32021)] and concrete computations.
scientific article; zbMATH DE number 5127965
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