Complete characterization of isolated homogeneous hypersurface singularities

Stephen Yau and Huaiqing Zuo

Vol. 273 (2015), No. 1, 213–224

DOI: 10.2140/pjm.2015.273.213

Abstract

Let $X$ be a nonsingular projective variety in ${ℂ\mathbb{P}}^{n-1}$. Then the cone over $X$ in ${ℂ}^n$ is an affine variety $V$ with an isolated singularity at the origin. It is a very natural and important question to ask when an affine variety with an isolated singularity at the origin is a cone over nonsingular projective variety.

This problem is very hard in general. In this paper we shall treat the hypersurface case. Given a function $f$ with an isolated singularity at the origin, we can ask whether $f$ is a weighted homogeneous polynomial or a homogeneous polynomial after a biholomorphic change of coordinates. The former question was answered in a celebrated 1971 paper by Saito. However, the latter question had remained open for 40 years until Xu and Yau solved it for $f$ with three variables. Recently, Yau and Zuo solved it for $f$ with up to six variables. However, the methods they used are hard to generalize. In this paper, we solve the latter question for general $n$ completely; i.e., we show that $f$ is a homogeneous polynomial after a biholomorphic change of coordinates if and only if $\mu =\tau ={\left(\nu -1\right)}^n$, where $\mu$, $\tau$ and $\nu$ are the Milnor number, Tjurina number and multiplicity of the singularity respectively. We also prove that there are at most ${\mu }^{1∕n}+1$ multiplicities within the same topological type of the isolated hypersurface singularity, while the famous Zariski multiplicity problem asserts that there is only one multiplicity.

Dedicated to Professor Michael Artin on the occasion of his 80th birthday

Keywords

homogeneous singularities, Milnor number, multiplicity

Mathematical Subject Classification 2010

Primary: 32S25

Secondary: 32S10

Milestones

Received: 8 February 2014

Accepted: 20 April 2014

Published: 6 December 2014