Non-existence of negative weight derivations on Jacobian algebras

Abstract

Let $S=\mathbb K[x_1,\cdots,x_n]$ be a polynomial ring in $n$ variables with positive integer weights over a field $\mathbb K$ of characteristic zero. Let $f\in S$ be a weighted homogeneous polynomial such that the Jacobian algebra $$A=\left.S \middle/ \left(\frac{\partial f}{\partial x_1},\cdots,\frac{\partial f}{\partial x_n}\right)\right.$$ is a finite-dimensional $\mathbb K$-vector space. We prove that $A$ has no nonzero negative weight derivation, which confirms a conjecture of the third author. An immediate consequence of this conjecture is the micro-local characterization of quasi-homogeneous singularities.