Example 16.3.22E548BFB3BF04

Example 16.3.22.E548BFB3BF04 We continue our favorite example example 5.20.6. Let $X=\mathbb {P}^1$ and $\omega $ be the Fubini–Study metric. We define $\varphi \in \mathrm{PSH}(X,\omega )$ as follows: For $z\in \mathbb {C}$, we let

while $\varphi (\infty )=2$. Note that $\varphi $ is $\omega $-psh on $X=\mathbb {P}^1$ by the gluing lemma 5.14.16. The singularity of $\varphi $ only occurs at $z=0$, close to which, $\varphi $ is approximately $-\log \left(-\log |z|^2\right)$. This type of singularity is therefore called the log-log type singularity.

We claim that

In particular, we find that $\varphi $ and $P_{\omega }[\varphi ]$ have different singularity types.

By theorem 16.3.15, in order to verify (16.3.23), it suffices to verify that

Here $\omega _{\varphi }^1$ is taken in the non-pluripolar sense. Since $\{ 0,\infty \} \subseteq \mathbb {P}^1$ is pluripolar, this reduces to showing that

where $\psi (z)=\varphi (z)+\log (|z|^2+1)$ and $\mu $ is the standard Lebesgue measure on $\mathbb {C}$.

Note that the Laplacian vanishes outside $\overline{B(0,0.7)}$ since $\psi (z)=2+\log |z|^2$ there, which is harmonic. Therefore,

It is an elementary exercise to see that the right-hand side is exactly equal to $1$. If you are familiar with toric geometry, this is more or less trivial since

Otherwise, just try to evaluate the integral using Green's identities. Therefore, (16.3.24) is proved and our assertion (16.3.23) follows.