My question is related to lemma 5.12 in ["Normal Subgroups in the Cremona Group"]. Let $h$ be a birational transformation of a projective surface $X$ and $[D']\in N^1(X)$ be an ample class. We define $[D]=[D']/\sqrt{[D']\cdot[D']}$ so that $[D] \in \mathbb{H}_{\overline{\mathcal{Z}}}$ where $\mathbb{H}_{\overline{\mathcal{Z}}}$ is the hyperboloid of one sheet derived from the (completed) Picard Manin space $\overline{\mathcal{Z}}(X)$ of $X$.

In the last line of the proof, the author stated that the induced isometry $h_\ast$ by $h$ on $\overline{\mathcal{Z}}(X)$ fixes the ample class $[D]$, and since $[D]$ is ample then $h$ is an automorphism of $X$.

My question how does $h_\ast$ fixing an ample class $[D]$ allow us to deduce that $h$ must be an automorphism on $X$? Any insight or resources proving the result will be greatly appreciated.