Corollary 21.13.43C521E79AA7F6

Proof. 2 $\iff $ 3. This follows from the definition of $Q_{\sigma }$ in (21.13.7).

1 $\iff $ 2. Let $\rho _1,\ldots ,\rho _ r$ be the rays of $\sigma $. Up to replacing $D$ by a translation, we may assume that $m_{\sigma }=0$. Hence, we may take $m_{\rho _ i}=0$ for all $i$.

Let $\pi \colon Z\rightarrow X$ be the blow-up of $X$ along $Y$. See [Page 132, CLS11] for the basic properties of the toric blow-up. Take the divisor $\pi ^*D$ on $Z$. We choose the pull-back metric $\pi ^*h$ on $\pi ^*L$. Then $F_{\pi ^*\theta }$ can be taken as $\pi ^*F_{\theta }$ by (21.13.23). It follows that $\Delta (\theta ,\varphi )=\Delta (\pi ^*\theta ,\pi ^*\varphi )$. On the other hand, the ray corresponding to the exceptional divisor $E$ is generated by $u_{\rho _1}+\cdots +u_{\rho _ r}$. Since $X$ is smooth, this vector is primitive.

Recall that the support function of $\pi ^*D$ is the same as the support function of $D$, see [Proposition 6.2.7, CLS11]. In particular, we can take the Cartier datum $m_{\rho }=m_{\sigma } \mod M(\rho )$, where $\rho $ is the ray corresponding to $E$.

It follows from corollary 21.13.42 and 5.16.17 that

Our assertion follows in view of (21.13.5). $\square$