Abelian subgroup of the automorphism group of $\mathbb C^n$

Question

Let $Aut(\mathbb C^n)$ be the automorphism group of $\mathbb C^n$, i.e., the group of all biholomorphic maps of $\mathbb C^n \to \mathbb C^n$. Suppose $T$ is a finitely dimensional torus which is a subgroup of $Aut(\mathbb C^n)$.

Can we prove that $T$, up to a conjugation, is contained in $GL(n, \mathbb C)$?