Fubini-Study on complex abelian varieties

Question

Let $A$ be a complex abelian variety. Then the Fubini-Study metric on $\mathbb{P}^N_{\mathbb{C}}$ restricts on to $A$ by pulling back along an embedding $A\hookrightarrow \mathbb{P}^N_{\mathbb{C}}$. I'd like to understand when an automorphism of $A$ gives an isometry with regard to the induced metric. The problem is that I don't have a good idea how I can express the metric on $A$, or rather on $A(\mathbb{C})^{an}\cong \mathbb{C}^g/L$.