A remark on lower bound of Milnor number and characterization of homogeneous hypersurface singularities

Abstract

Let f : (Cn+1,0) → (C, 0) be a holomorphic germ defining an isolated hypersurface singularity V at the origin. Let μ and ν and p g be the Milnor number, multiplicity and geometric genus of (V, 0), respectively. We conjecture that μ ≥ (ν - 1)n+1 and the equality holds if and only if f is a semi-homogeneous function. We prove that this inequality holds for n = 1, and also for n = 2 or 3 with additional assumption that f is a quasihomogeneous function. For n = 1, if V has at most two irreducible branches at the origin, or if f is a quasi-homogeneous function, then μ = (ν - 1)2 if and only if f is a homogeneous polynomial. For n = 2, if f is a quasi-homogeneous function, then μ = (ν -1)3 iff 6pg = ν(ν - 1)(ν - 2) iff f is a homogeneous polynomial after biholomorphic change of variables. For n = 3, if f is a quasi-homogeneous function, then μ = (ν - 1)4 iff 24p g = ν(ν - 1)(ν - 3)(ν - 3) iff f is a homogeneous polynomial after biholomorphic change of variables.