Pure and Applied Mathematics Quarterly

Volume 12 (2016)

Number 1

Special Issue: In Honor of Eduard Looijenga, Part 2 of 3

Guest Editor: Gerard van der Geer

Sharp upper estimate conjecture for the Yau number of a weighted homogeneous isolated hypersurface singularity

Pages: 165 – 181

DOI: http://dx.doi.org/10.4310/PAMQ.2016.v12.n1.a6

Authors

Stephen S.-T. Yau (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Huai Qing Zuo (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Abstract

Let $V$ be a hypersurface with an isolated singularity at the origin defined by the function of $f : (\mathbb{C}^n, 0) \to (\mathbb{C}, 0)$. Let $L(V)$ be the Lie algebra of derivations of the moduli algebra $A(V) := \mathbb{C} \{ x_1, \dotsm , x_n \} / (f, \partial f / \partial x_1, \dotsm , \partial f / \partial x_n)$. It is known that $L(V)$ is a finite dimensional solvable Lie algebra ( [Ya1], [Ya2]). $L(V)$ is called the Yau algebra of $V$ in [Yu] and [Khi] in order to distinguish from Lie algebras of other types appearing in singularity theory ([AVZ], [AM], [BY]). $\dim L(V)$ is called Yau number in [EK], [Khi]. This number is an analytic invariant. In case $V$ is a weighted homogeneous singularity, it is a natural question whether Yau number can be bounded by a number which depends only on the weight. In this paper we formulate a sharp upper estimate conjecture for the Yau number of weighted homogeneous isolated hypersurface singularities.We prove this conjecture for a large class of singularities.

Keywords

Yau number, isolated hypersurface singularity, derivation Lie algebra

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