PomeCorollary 16.4.8.953C40C3B9B8 Let $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$ be a model potential. Let $\omega $ be a Kähler form on $X$. Then
\[ \varphi =\inf _{\epsilon >0} P_{\theta +\epsilon \omega }[\varphi ]. \]
Corollary 16.4.8953C40C3B9B8
Proof. Set
\[ \eta \coloneqq \inf _{\epsilon >0}P_{\theta +\epsilon \omega }[\varphi ]. \]
Since $\varphi $ is model, $\varphi =P_{\theta }[\varphi ]\leq P_{\theta +\epsilon \omega }[\varphi ]$ for every $\epsilon >0$, hence $\varphi \leq \eta \leq 0$.
Next we prove the reverse direction. Applying proposition 16.4.4 to the decreasing net $(P_{\theta +\epsilon \omega }[\varphi ])_{\epsilon >0}$ and using proposition 16.3.10, we get
\[ P_\theta [\eta ]=\eta ,\quad \int _ X\theta _\eta ^ n=\int _ X\theta _\varphi ^ n. \]
Thus $\eta $ is a candidate in the characterization theorem 16.3.15 of $P_\theta [\varphi ]$. Hence $\eta \leq P_\theta [\varphi ]=\varphi $. $\square$
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