PomeExample 21.12.22.2B1E84EAAE13 Now we can give an example of $\varphi ,\psi \in \mathrm{PSH}_{\mathrm{tor}}(X,\omega )$ with $\varphi \land \psi \equiv -\infty $.
We take $P=[0,1]$ so that $X=\mathbb {P}^1$ and $\omega $ is the Fubini–Study metric. Let $\varphi \in \mathrm{PSH}(X,\omega )$ be such that
\[ \varphi (z)=\log \frac{|z|^2}{|z|^2+1} \]
for $z\in \mathbb {C}$. We have computed that $G_{\varphi }$ in example 21.12.16:
\[ G_{\varphi }(m)= \begin{cases} 0,& \textup{if } m=1,\\ \infty ,& \textup{otherwise}. \end{cases} \]
Now we define $\psi \in \mathrm{PSH}_{\mathrm{tor}}(X,\omega )$ as the unique function such that
\[ \psi (z)=\log \frac{1}{|z|^2+1} \]
for $z\in \mathbb {C}$. Then a similar computation shows that
\[ G_{\psi }(m)= \begin{cases} 0,& \textup{if } m=0,\\ \infty ,& \textup{otherwise}. \end{cases} \]
Now we claim that $\varphi \land \psi \equiv -\infty $. Otherwise, we would have
\[ G_{\varphi \land \psi }=G_{\varphi }\lor G_{\psi }\equiv \infty , \]
which is not proper.
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