#### Vol. 314, No. 2, 2021

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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
##### 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 ${\mathsc{ℒ}}_{k}\left(V\right)$ is defined to be the Lie algebra of derivations of the new $k$-th local algebra ${M}_{k}\left(V\right)$, where ${M}_{k}\left(V\right):={\mathsc{𝒪}}_{n}∕\left(f+{J}_{1}\left(f\right)+\cdots +{J}_{k}\left(f\right)\right)$. 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 ${\mathsc{ℒ}}_{3}\left(V\right)$ 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
Revised: 30 May 2021
Accepted: 16 July 2021
Published: 10 November 2021
##### Authors
 Naveed Hussain Guangzhou Huashang College Guangzhou China Stephen S.-T. Yau Tsinghua University Beijing China Yanqi Lake Beijing Institute of Mathematical Sciences and Applications Beijing China Huaiqing Zuo Tsinghua University Beijing China