Example 16.8.18C7A2AD00BFF3

Example 16.8.18.C7A2AD00BFF3 Let us continue our example example 16.3.22, where $X=\mathbb {P}^1$, $\omega $ is the Fubini–Study metric and $\varphi \in \mathrm{PSH}(X,\omega )$ has log-log singularity at $0$. We have shown that $P_{\omega }[\varphi ]=0$ in (16.3.23), so $\varphi \sim _{P} 0$ and hence $\varphi \sim _{\mathcal{I}} 0$. In particular, $P$-equivalence is not equivalent to the equivalence of singularity types.

On the other hand, consider a potential $\psi \in \mathrm{PSH}(X,\omega )$ with log singularity at $0$, as in example 5.20.12. We know that $\nu (\psi ,0)=1$ from the explicit expression (5.20.13). So $\psi \not\sim _{\mathcal{I}} 0$ and hence $\psi \not\sim _{P} 0$.

Moreover, $\psi \preceq _{P} \varphi $ and hence $\psi \preceq _{\mathcal{I}}\varphi $.