Inequality Conjectures on derivations of Local $k$-th Hessain algebras associated to isolated hypersurface singularities

Naveed Hussain Stephen S.-T. Yau Tsinghua University Huaiqing Zuo Tsinghua University

Algebraic Geometry mathscidoc:2010.45001

Let $(V, 0)$ be an isolated hypersurface singularity. We introduce a series of new derivation Lie algebras $L_{k}(V)$ associated to $(V,0)$. Its dimension is denoted as $\lambda_{k}(V)$. The $L_{k}(V)$ is a generalization of the Yau algebra $L(V)$ and $L_{0}(V)=L(V)$. These numbers $\lambda_{k}(V)$ are new numerical analytic invariants of an isolated hypersurface singularity. In this article we compute $L_1(V)$ for fewnomial isolated singularities (Binomial, Trinomial) and obtain the formulas of $L_{1}(V)$. We also formulate a sharp upper estimate conjecture for the $L_k(V)$ of weighted homogeneous isolated hypersurface singularities and we prove this conjecture for large class of singularities. Furthermore, we formulate another conjecture: $\cdots >\lambda^{(i+1)}(V) > \lambda^i(V) \cdots >\lambda^0(V) \geq \lambda_l (V),\; l=1,2,\cdots,n$. We partially prove it for binomial and trinomial singularities.
isolated hypersurface singularity, Lie algebra, moduli algebra
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@inproceedings{naveedinequality,
  title={ Inequality Conjectures on derivations of Local $k$-th Hessain algebras associated to isolated hypersurface   singularities},
  author={Naveed Hussain, Stephen S.-T. Yau, and Huaiqing Zuo},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20201027223613663537720},
}
Naveed Hussain, Stephen S.-T. Yau, and Huaiqing Zuo. Inequality Conjectures on derivations of Local $k$-th Hessain algebras associated to isolated hypersurface singularities. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20201027223613663537720.
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