Corollary 21.13.60396E3B9D09BD

Proof. 1 $\iff $ 2. This follows from the $\mathcal{I}$-goodness of $\varphi $ and $\psi $, see example 18.1.56.

1 $\iff $ 3. When $\varphi ,\psi \in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )_{>0}$, this follows from proposition 21.13.58.

In general, fix an ample toric-invariant divisor $H$ on $X$ and a toric-invariant Hermitian metric $h_ H$ on $\mathcal{O}_ X(H)$ such that $\omega _ H\coloneqq \mathrm{dd}^{\mathrm{c}}h_ H$ is a Kähler form. We perturb $L$ to $L+m^{-1}H$ for $m\in \mathbb {Z}_{>0}$, then thanks to 21.11.47, we find that

Then since $\varphi ,\psi \in \mathrm{PSH}_{\mathrm{tor}}(X,\theta +m^{-1}\omega _ H)_{>0}$, we know that $\varphi \preceq _ P \psi $ if and only if

for any $m\in \mathbb {Z}_{>0}$. The latter condition is equivalent to $\Delta (\theta ,\varphi )\subseteq \Delta (\theta ,\psi )$. $\square$