A $\bar{\partial}\partial$-Poincaré lemma for forms near an isolated complex singularity

Let $X$ be an analytic subvariety of complex Euclidean space with isolated singularity at the origin, and let $\eta$ be a smooth form of type $(1.1)$ defined on $X \setminus\{0\}$. The main result of this note is a criterion for solubility of the equation $\bar{\partial}\partial u = \eta$. This implies a criterion for triviality of a Hermitian- holomorphic line bundle $(L,h)\to X\setminus\{0\}$ in a neighbourhood of the origin.