Example 7.29 . [02XS]

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Example 7.29.

Let ${\mathcal{O}}(1)$ be the universal line bundle of $\mathbb{P}^{n}$ . The Fubini-Study metric of ${\mathcal{O}}(1)^{{\text{\rm an}}}$ corresponds to the case of the standard simplex, $\ell_{i}(x)=x_{i}$ , $i=1,\dots,n$ and $\ell_{n+1}(x)=1-\sum_{i=1}^{n}x_{i}$ and the choice $c_{i}=1/2$ for all $i$ . Hence we recover from (7.28) the well known expression for the height of $\mathbb{P}^{n}$ with respect to the Fubini-Study metric in [BGS94, Lemma 3.3.1]:

\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(\mathbb{P}^{n})=\frac{n+1}{2}\sum_{j=2}^{n+1}\frac{1}{j}.