A characterization and solvability of quasihomogeneous singularities

Guorui Ma, Stephen S.-T. Yau, Qiwei Zhu and Huaiqing Zuo

Vol. 329 (2024), No. 1, 121–146

DOI: 10.2140/pjm.2024.329.121

Abstract

Let $(V,0)$ be an isolated hypersurface singularity defined by the holomorphic function $f:({ℂ}^{n+1},0)\to (ℂ,0)$. A local $k$-th ($0\le k\le n+1$) Hessian algebra $H_k(V)$ of isolated hypersurface singularity $(V,0)$ is a finite-dimensional $ℂ$-algebra and it depends only on the isomorphism class of the germ $(V,0)$. It is a natural question to ask for a necessary and sufficient condition for a complex analytic isolated hypersurface singularity to be quasihomogeneous in terms of its local $k$-th Hessian algebra $H_k(f)$. Xu and Yau proved that $(V,0)$ admits a quasihomogeneous structure if and only if $H_0(f)$ is isomorphic to a finite-dimensional nonnegatively graded algebra in the early 1980s. In this paper, on the one hand, we generalize Xu and Yau’s result to $H_{n+1}(f)$. On the other hand, a new series of finite-dimensional Lie algebras $L_k(V)$ (resp. $L^k(V)$) was defined to be the Lie algebra of derivations of the $k$-th ($0\le k\le n+1$) Hessian algebra $H_k(V)$ (resp. $A^k(V):={\mathcal{𝒪}}_{n+1}∕(f,m^k{J}_f)$) and is finite-dimensional. We prove that $(V,0)$ is quasihomogeneous singularity if $L_{n+1}(V)$ (resp. $L^k(V):=\text{ Der}(A^k(V))$) satisfies certain conditions. Moreover, we investigate whether the Lie algebras $L_k(V)$ (resp. $L^k(V)$) are solvable.

Keywords

solvability of derivation Lie algebra, isolated quasihomogeneous singularities.

Mathematical Subject Classification

Primary: 14B05, 32S05

Milestones

Received: 9 August 2023

Revised: 15 January 2024

Accepted: 14 April 2024

Published: 12 June 2024