Example 5.20.12CC1C409253F3

Example 5.20.12.CC1C409253F3 We continue with our example example 5.20.6. Let $X=\mathbb {P}^1$ and $L=\mathcal{O}_{\mathbb {P}^1}(1)$. Let $h^0$ denote the Fubini–Study metric on $L$ as defined in example 5.20.6. Note that we have changed the notation from $h$ to $h^0$. Let $\omega =\mathrm{dd}^{\mathrm{c}}h^0$.

We construct $\varphi \in \mathrm{PSH}(X,\omega )$ as follows: On $\mathbb {C}$, define

Then $\varphi \in \mathrm{PSH}(\mathbb {C},\omega |_{\mathbb {C}})$ by (5.20.7). Setting $\varphi (\infty )=0$, we can easily verify that $\varphi \in \mathrm{PSH}(\mathbb {P}^1,\omega )$.⁵

We then get a plurisubharmonic metric $h=h^0\exp (-\varphi )$. To be more explicit, the fiber metric $h$ at $0$ is singular, $h_{\infty }=h^0_{\infty }$, while for $z\in \mathbb {C}\setminus \{ 0\} $ and $\lambda \in [z]^{\vee }$, we have