Say $V$ is 1-Gorenstein iff one of the following equivalent conditions holds:

1. (1)

   Every $x\in V$ has an open neighborhood $U$ such that $U^{reg}$ carries an holomorphic $n$-form with an empty zero divisor.

2. (2)

   $\omega_{V}$ is a rank one locally free sheaf.

3. (3)

   Every $x\in V$ has an open neighborhood $U$ such that $\omega_{U^{reg}}$ is isomorphic to $\mathcal{O}_{V^{reg}}|_{U}$.

A local section of $\omega_{V}$ defining an holomorphic $n$-form without zeroes on $V^{reg}$ will be called a local generator of $\omega_{V}$. If furthermore $V$ is Cohen-Macaulay, $V$ is said to be Gorenstein.

Say $V$ is $\mathbb{Q}$-Gorenstein iff one of the following equivalent conditions is satisfied:

1. (1)

   Every $x\in V$ has an open neighborhood $U$ such that $U^{reg}$ carries an holomorphic pluricanonical form with an empty zero divisor.

2. (2)

   For every $x\in V$, there exists $N_{x}\in\mathbb{N}$ and an open neighborhood $U$ of $x$ such that $\omega^{[N_{x}]}_{U}$ is a rank one locally free sheaf.

3. (3)

   For every $x\in V$ there is $N_{x}\in\mathbb{N}$ and an open neighborhood $U$ of $x$ such that $\omega^{N_{x}}_{U^{reg}}$ is isomorphic to $\mathcal{O}_{V^{reg}}|_{U}$.

A local section of $\omega^{[N]}_{V}$ defining an holomorphic pluricanonical form without zeroes on $V^{reg}$ will be called a local generator of $\omega^{[N]}_{V}$.

For every $x\in V$, the smallest $N_{x}$ fulfilling condition 3 near $x$ is called the local index of $V$ at $x$. The l.c.m. of all local indices, if finite, is called the index of $V$.