AI 中文总结
该研究推导孤立复超曲面奇点的广义模代数相关性质，证明其Yau代数维数差由Hessian余秩决定，验证了Chen等人的猜想，还通过同调方法完善了相关定理的证明。
AI 中文摘要
设$(V,0)$是由$f\in\mathbb{C}\{x_1,\ldots,x_n\}$定义的孤立复超曲面奇点。令$A(V)$为模（Tjurina）代数，$A^*(V)$为广义模代数，且令$L(V)=\mathrm{Der}_{\mathbb{C}}(A(V),A(V))$、$L^*(V)=\mathrm{Der}_{\mathbb{C}}(A^*(V),A^*(V))$分别为$V$的Yau代数和新Yau代数。记$\lambda(V)=\dim_{\mathbb{C}}L(V)$，$\lambda^*(V)=\dim_{\mathbb{C}}L^*(V)$。我们证明这两个维数的差由Hessian余秩决定：在Morse情形下，广义模代数为零，因此两个导子李代数均为零；当$\mathrm{corank}(\mathrm{Hess}(f)(0))=1$时，$\lambda^*(V)=\lambda(V)-1$，否则$\lambda^*(V)=\lambda(V)$；特别地，当$n\geq2$且$\mathrm{mult}(f)\geq3$时，$\lambda^*(V)=\lambda(V)$。这证明了文献[ChenHussainYauZuo2020]提出的猜想1.1。我们构造了一个正合序列，其两端项均为Milnor代数的基座，该同调方法揭示了维数计数背后的详细代数结构；我们还给出了一个替代证明，进一步推进了文献[ChenHussainYauZuo2020]中定理C的原始论证，一般情形通过结合Saito的非拟齐次情形准则得到。
英文摘要
Let $(V,0)$ be an isolated complex hypersurface singularity defined by $f\in\mathbb{C}\{x_1,\ldots,x_n\}$. Let $A(V)$ be the moduli(Tjurina) algebra, let $A^*(V)$ be the generalized moduli algebra, and let $$ L(V)=\mathrm{Der}_{\mathbb{C}}(A(V),A(V)),\qquad L^*(V)=\mathrm{Der}_{\mathbb{C}}(A^*(V),A^*(V)) $$ be the Yau algebra and the new Yau algebra of $V$, respectively. Write $λ(V)=\dim_{\mathbb{C}}L(V)$ and $λ^*(V)=\dim_{\mathbb{C}}L^*(V)$. We prove that the difference between these two dimensions is determined by the Hessian corank. In the Morse case, the generalized moduli algebra is zero, and hence both derivation Lie algebras are zero. We have $λ^*(V)=λ(V)-1$ if $\mathrm{corank}(\mathrm{Hess}(f)(0))=1$, and $λ^*(V)=λ(V)$ otherwise. In particular, $λ^*(V)=λ(V)$ when $n\ge2$ and $\mathrm{mult}(f)\ge3$. This proves Conjecture~1.1 proposed in \cite{ChenHussainYauZuo2020}. We construct an exact sequence whose two end terms are copies of the socle of the Milnor algebra. This homological approach reveals the detailed algebraic structure underlying the dimension count. We also give an alternative proof that pushes the original arguments in the proof of \cite[Theorem~C]{ChenHussainYauZuo2020} further. The general case is obtained by combining Saito's criterion for the non-quasi-homogeneous case.
