$k$TH YAU NUMBER OF ISOLATED HYPERSURFACE SINGULARITIES AND AN INEQUALITY CONJECTURE. (30th April 2019)

• Record Type: Journal Article
• Authors: HUSSAIN, NAVEED; YAU, STEPHEN S.-T.; ZUO, HUAIQING
• Is Part Of: Journal of the Australian Mathematical Society. Volume 110:Number 1(2021)
• Page Start: 94
• Page End: 118
• Publication Date: 2019-04-30
• Subjects: 14B05, -- 32S05; derivation, -- Lie algebra, -- isolated singularity, -- Yau algebra; Mathematics -- Periodicals; Statistics -- Periodicals; Mathematical statistics -- Periodicals; 510.5
• DOI: 10.1017/S1446788719000132
• Languages: English
• ISSNs: 1446-7887

Abstract

Abstract: 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.