Let $G \subset SL(n,\mathbb{C})$ be a cyclic subgroup of finite order,
Is it true that $\mathbb{C}^n /G$ is toric ? If not then when it is ?



When a quotient singularity is toric?
Yes, it is true. Let $G \cong \mathbb{Z}/m$ act by $\mathrm{diag}(\zeta^{a\_1}, \zeta^{a\_2}, \ldots, \zeta^{a\_n})$, where $\zeta$ is a primitive $n$-th root of unity. Let $S$ be the semigroup $\{ (b\_1, \ldots, b\_n) \in \mathbb{Z}\_{\geq 0}^n : \sum a\_i b\_i \equiv 0 \mod m \}$. Then the quotient is Spec of the semigroup ring $\mathbb{C}[S]$. Since the semigroup is torsion free, finitely generated and saturated, the corresponding Spec is an affine toric variety.


