TH YAU NUMBER OF ISOLATED HYPERSURFACE SINGULARITIES AND AN INEQUALITY CONJECTURE

Let $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)$ . The Yau algebra $L(V)$ is defined to be the Lie algebra of derivations of the moduli algebra $A(V):={\mathcal{O}}_{n}/(f,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{1},\ldots ,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{n})$ , that is, $L(V)=\text{Der}(A(V),A(V))$ . It is known that $L(V)$ is finite dimensional and its dimension $\unicode[STIX]{x1D706}(V)$ is called the Yau number. We introduce a new series of Lie algebras, that is, $k$ th Yau algebras $L^{k}(V)$ , $k\geq 0$ , which are a generalization of the Yau algebra. The algebra $L^{k}(V)$ is defined to be the Lie algebra of derivations of the $k$ th moduli algebra $A^{k}(V):={\mathcal{O}}_{n}/(f,m^{k}J(f)),k\geq 0$ , that is, $L^{k}(V)=\text{Der}(A^{k}(V),A^{k}(V))$ , where $m$ is the maximal ideal of ${\mathcal{O}}_{n}$ . The $k$ th Yau number is the dimension of $L^{k}(V)$ , which we denote by $\unicode[STIX]{x1D706}^{k}(V)$ . In particular, $L^{0}(V)$ is exactly the Yau algebra, that is, $L^{0}(V)=L(V),\unicode[STIX]{x1D706}^{0}(V)=\unicode[STIX]{x1D706}(V)$ . These numbers $\unicode[STIX]{x1D706}^{k}(V)$ are new numerical analytic invariants of singularities. In this paper we formulate a conjecture that $\unicode[STIX]{x1D706}^{(k+1)}(V)>\unicode[STIX]{x1D706}^{k}(V),k\geq 0.$ We prove this conjecture for a large class of singularities.

ON THE DERIVATION LIE ALGEBRAS OF FEWNOMIAL SINGULARITIES

Let $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)$. The Yau algebra, $L(V)$, is the Lie algebra of derivations of the moduli algebra of $V$. It is a finite-dimensional solvable algebra and its dimension $\unicode[STIX]{x1D706}(V)$ is the Yau number. Fewnomial singularities are those which can be defined by an $n$-nomial in $n$ indeterminates. Yau and Zuo [‘A sharp upper estimate conjecture for the Yau number of weighted homogeneous isolated hypersurface singularity’, Pure Appl. Math. Q.12(1) (2016), 165–181] conjectured a bound for the Yau number and proved that this conjecture holds for binomial isolated hypersurface singularities. In this paper, we verify this conjecture for weighted homogeneous fewnomial surface singularities.