Example 5.20.60D9E7B0EE3DC

Example 5.20.6.0D9E7B0EE3DC Let $X=\mathbb {P}^1$ and $L=\mathcal{O}_{\mathbb {P}^1}(1)$. The homogeneous coordinates on $\mathbb {P}^1$ will be denoted by $[X_0:X_1]$. At a point $x=[X_0:X_1]\in \mathbb {P}^1$, the fiber $L_ x$ is identified with the dual of $[x]$, where $[x]\subseteq \mathbb {C}^2$ is the line represented by $x$.

In order to introduce the Hermitian metric $h$ on $L$, we fix the standard Hermitian norm $\| \bullet \| $ on $\mathbb {C}^2$. Then given $\lambda \in L_ x=[x]^{\vee }$, we introduce

where $\tilde{x}$ is an arbitrary non-zero element in $[x]$. The reader can easily verify that $h$ is indeed a Hermitian metric on $L$. The Hermitian metric $h$ is known as the Fubini–Study metric.

A holomorphic section $s\in \mathrm{H}^0(X,L)$ can be formally identified with a linear form $a_0X_0+a_1X_1$: At $x\in X$, the corresponding linear form on $[x]$ is given by sending $(X_0,X_1)$ to $a_0X_0+a_1X_1$.

Next we compute $\mathrm{dd}^{\mathrm{c}}h=c_1(L,h)$. For this purpose, we cover $\mathbb {P}^1$ by $\mathbb {C}=\mathbb {P}^1\setminus \{ \infty \} $ and $\mathbb {P}^1\setminus \{ 0\} $. Both are holomorphic coordinate charts with coordinate function $z=X_0/X_1$ and $z^{-1}=X_1/X_0$ respectively.

We claim that on $\mathbb {C}$,

In fact, let $t$ be the nowhere vanishing section of $L$ on $\mathbb {C}$ corresponding to $X_1$. Then for $z\in \mathbb {C}$, we have an obvious lift $(z,1)\in [z]$, so

So (5.20.7) follows.

In order to obtain a non-trivial case of the Lelong–Poincaré formula, we need to consider a section which vanishes at some points in $\mathbb {C}$. Let $s$ be the holomorphic section of $L$ corresponding to $X_0$. Then

for any $z\in \mathbb {C}$ using the same argument as above. Therefore, we find that restricted to $\mathbb {C}$, we have

where $f(z)=\log |z|^2$. So the Lelong–Poincaré formula (5.20.5) is verified in this case.

The Kähler form $\mathrm{dd}^{\mathrm{c}}h$ on $\mathbb {P}^1$ is also known as the Fubini–Study metric.