Example 2.32 . [02JE]

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Example 2.32.

Let $X=\mathbb{P}^{n}$ be the projective space over $\mathbb{C}$ and $L=\mathcal{O}(1)$ . The canonical metric of $\mathcal{O}(1)^{{\text{\rm an}}}$ is the metric given, for $p=(p_{0}:\dots:p_{n})\in\mathbb{P}^{n}(\mathbb{C})$ , by

\|s(p)\|_{\operatorname{can}}=\frac{|\rho_{s}(p_{0},\dots,p_{n})|}{\max_{i}\{|p_{i}|\}},

for any rational section $s$ of $L$ defined at $p$ and the homogeneous rational function $\rho_{s}\in\mathbb{C}(x_{0},\dots,x_{n})$ associated to $s$ .

This is an approachable metric. Indeed, consider the $m$ -power map $[m]:\mathbb{P}^{n}\to\mathbb{P}^{n}$ defined as $[m](p_{0}:\dots:p_{n})=(p^{m}_{0}:\dots:p^{m}_{n})$ . The $m$ -th root of the inverse image by $[m]$ of the Fubini-Study metric of $\mathcal{O}(1)^{{\text{\rm an}}}$ is the semipositive smooth metric on $L^{{\text{\rm an}}}$ given by

\|s(p)\|_{m}=\frac{|s(p_{0},\dots,p_{n})|}{(\sum_{i}|p_{i}|^{2m})^{1/2m}}.

The family of metrics obtained varying $m$ converges uniformly to the canonical metric.