Durfee conjecture and coordinate free characterization of homogeneous singularities (Q688329)

This article proves a sharp version of Durfee's conjecture in the case of weighted homogeneous polynomials in 3 variables: If \((V,0)\) is an isolated singularity defined in \(\mathbb{C}^ 3\) by a weighted homogeneous polynomial, then the inequality \[ \mu-\nu+1 \geq 6Pg \] is valid, with equality iff we are in the homogeneous case. Here \(\mu\) is the Milnor number, \(\nu\) the multiplicity and \(Pg\) the geometric genus of \((V,0)\). The signature \(\sigma\) of the Milnor fiber satisfies \[ \sigma \leq-{\mu \over 3}-{2 \over 3} (\nu-1) \] (which implies \(\sigma \leq 0)\). As a corollary, this gives, using K. Saito's classical characterization of weighted homogeneous isolated hypersurfaces singularities, a characterization of (biholomorphic) homogeneous isolated hypersurfaces singularities in \(\mathbb{C}^ 3\).