Derivations of Generalized Moduli Algebras of Isolated Hypersurface Singularities

Abstract

Let $(V,0)$ be an isolated complex hypersurface singularity defined by $f\in \mathbb{C}\{x_1,\dots ,x_n\}$. Let $A(V)$ be the moduli(Tjurina) algebra, let $A^{\ast }(V)$ be the generalized moduli algebra, and let $$L(V)={\mathrm{Der}}_{\mathbb{C}}(A(V),A(V)),L^{\ast }(V)={\mathrm{Der}}_{\mathbb{C}}(A^{\ast }(V),A^{\ast }(V))$$ be the Yau algebra and the new Yau algebra of $V$, respectively. Write $\lambda (V)={\dim }_{\mathbb{C}}L(V)$ and ${\lambda }^{\ast }(V)={\dim }_{\mathbb{C}}{L}^{\ast }(V)$. 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 ${\lambda }^{\ast }(V)=\lambda (V)-1$ if $\mathrm{corank}(\mathrm{Hess}(f)(0))=1$, and ${\lambda }^{\ast }(V)=\lambda (V)$ otherwise. In particular, ${\lambda }^{\ast }(V)=\lambda (V)$ when $n\ge 2$ and $\mathrm{mult}(f)\ge 3$. 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.