Example 10.6.74CE1D9D7D803

Example 10.6.7.4CE1D9D7D803 Let $\mathcal O_{\mathbb P^ N}(1)$ carry the Fubini–Study metric $h_{\mathrm{FS}}$ of example 10.6.6, dual to the Euclidean norm on the tautological line. On the chart $U_0=\{ [1:z_1:\cdots :z_ N]\} $ the tautological line is spanned by $(1,z)$, so the frame $X_0=e_0^*$ has squared norm $|X_0(1,z)|^2/|(1,z)|^2=(1+|z|^2)^{-1}$, and the curvature is

The matrix of $\partial _ j\partial _{\bar k}\log (1+|z|^2)$ is $\bigl ((1+|z|^2)\delta _{jk}-\bar z_ jz_ k\bigr )/(1+|z|^2)^2$, which is positive definite by the Cauchy–Schwarz inequality $|\sum _ j\bar z_ j\xi _ j|^2\leq |z|^2|\xi |^2$. The same computation in the chart $U_ i$, with the frame $X_ i$, shows that $\omega _{\mathrm{FS}}$ is a Kähler form on $\mathbb P^ N$; thus $\mathbb P^ N$ is a compact Kähler manifold. On $\mathbb P^1$, integration in polar coordinates gives $\int _{\mathbb P^1}\omega _{\mathrm{FS}}=\frac1\pi \int _{\mathbb C}(1+|z|^2)^{-2}\, \mathrm dV=1$. The function $\log |X_1|_{\mathrm{FS}}^2=\log \bigl (|z|^2/(1+|z|^2)\bigr )$, which vanishes at the point at infinity, is the weight of a singular plurisubharmonic metric whose Chern current is the Dirac current $[0]$ at the point $[1:0]$, by (10.6.5).