Say $V$ has only log-terminal singularities iff $V$ is $\mathbb{Q}$-Gorenstein, of finite index $N$ and the following holds: let $\pi:X\to V$ be a log-resolution and let $\alpha$ be a local generator of $\omega_{V}^{[N]}$: then the pole along any component $E$ of $exc(\pi)$ of the meromorphic $N$-canonical form $\pi^{*}\alpha$ on $X$ is of order $\leq N-1$.

When $V$ is algebraic an equivalent formulation is: let $\pi:X\to V$ be a log-resolution. Then $K_{X}\cong\pi^{*}K_{V}+\displaystyle\sum_{E}a_{E}E$ with $a_{E}>-1$.