Corollary 21.13.7986EC82D6C082

Proof. If $\nu (\varphi ,Y)>0$, then by corollary 21.13.43, $\Delta (\theta ,\varphi )\cap Q_\sigma =\varnothing $. By convention the right-hand side of (21.13.80) is $0$, and the trace is not defined, so the left-hand side is $0$.

Assume now that $\nu (\varphi ,Y)=0$. Take a toric-invariant ample line bundle $H$ on $X$ and a toric-invariant Kähler metric $\omega \in c_1(H)$. For $\epsilon \in \mathbb {Q}_{>0}$, write $Q_{\sigma ,\epsilon }$ for the face corresponding to $\sigma $ for the divisor $D+\epsilon H$, with the same convention on the Cartier datum along $\sigma $. Since $\theta _{\varphi }+\epsilon \omega $ is a toric Kähler current and $\nu (\varphi ,Y)=0$, its trace along $Y$ is defined and has positive volume. Hence theorem 21.13.75 gives

By 21.11.47,

We choose the Cartier datum of $D+\epsilon H$ along $\sigma $ to be the sum of the chosen datum of $D$ and $\epsilon $ times the chosen datum of $H$. With this convention, the face of $P_ D+\epsilon P_ H$ corresponding to $\sigma $ is $Q_\sigma +\epsilon Q_\sigma ^ H$. In particular,

where $Q^ H_{\sigma }$ denotes the corresponding face of the polytope of $H$. Indeed, the defining inequalities along the rays of $\sigma $ are non-negative on both summands, so equality for their sum forces equality on each summand. By example 20.2.17, the left-hand side converges to

with the convention that this limit is $0$ when the trace in the limiting class is not defined. The right-hand side converges to

Hence (21.13.80) follows. $\square$