Derivation Lie algebras of new $k$-th local algebras of isolated hypersurface singularities

Naveed Hussain, Stephen S.-T. Yau and Huaiqing Zuo

Vol. 314 (2021), No. 2, 311–331

DOI: 10.2140/pjm.2021.314.311

Abstract

Let $\left(V,0\right)=\left\{\left(z_1,\dots ,z_n\right)\in {ℂ}^n:f\left(z_1,\dots ,z_n\right)=0\right\}$ be an isolated hypersurface singularity with $mult\left(f\right)=m$. Let $J_k\left(f\right)$ be the ideal generated by all $k$-th order partial derivative of $f$. For $1\le k\le m-1$, the new object ${\mathcal{ℒ}}_k\left(V\right)$ is defined to be the Lie algebra of derivations of the new $k$-th local algebra $M_k\left(V\right)$, where $M_k\left(V\right):={\mathcal{𝒪}}_n∕\left(f+J_1\left(f\right)+\cdots +J_k\left(f\right)\right)$. Its dimension is denoted as ${\delta }_k\left(V\right)$. This number ${\delta }_k\left(V\right)$ is a new numerical analytic invariant. We compute ${\mathcal{ℒ}}_3\left(V\right)$ for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of ${\delta }_3\left(V\right)$. We also formulate a sharp upper estimate conjecture for the ${\delta }_k\left(V\right)$ of weighted homogeneous isolated hypersurface singularities and verify this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture: ${\delta }_{\left(k+1\right)}\left(V\right)<{\delta }_k\left(V\right)$, $k\ge 1$ and verify it for low-dimensional fewnomial singularities.

Keywords

isolated hypersurface singularity, Lie algebra, local algebra

Mathematical Subject Classification

Primary: 14B05, 32S05

Milestones

Received: 15 February 2021

Revised: 30 May 2021

Accepted: 16 July 2021

Published: 10 November 2021