Existence of meromorphic 2-forms over normal surface singularities

Question

Let $(X,o)$ be an isolated normal surface singularity. Denote by $U:=X\backslash \{o\}$. I am looking for conditions on $(X,o)$ under which there exists a holomorphic section $\omega \in H^0(U, \Omega^2_U)$ which does not vanish anywhere on $U$. If I understand correctly, this holds if $(X,o)$ is Gorenstein. Is there any other example?

Answer 1

This is equivalent to the Gorenstein condition. If $\omega_X$ denotes the dualizing module, then $\omega_X|U=\Omega^2_U$. So, you get a nowhere vanishing section of $\omega_X|U$. Since $\omega_X$ has depth 2, this section extends to a nowhere vanishing section of $\omega_X$ and thus $(X,o)$ is Gorenstein.