Example 10.7.13E7DB735CB52F

Example 10.7.13.E7DB735CB52F On $\mathbb P^1$ take $L=\mathcal O(1)$ with its Fubini–Study metric, $T$ trivial, $\varphi =0$ and $\mathrm dV=\omega _{\mathrm{FS}}$. A section of $L^ k$ is $f(z)X_0^ k$ on the chart $[1:z]$ with $f$ entire, and in the chart at infinity, with $\zeta =1/z$, it equals $\zeta ^ kf(1/\zeta )X_1^ k$; holomorphicity at $\zeta =0$ and Cauchy's estimates show that $f$ is a polynomial of degree at most $k$. Integration over circles makes the monomials orthogonal, and with $t=|z|^2$,

the last equality follows from the substitution $u=t/(1+t)$ and repeated integration by parts. Thus the sections $\sqrt{(k+1)\binom {k}{j}}X_0^{k-j}X_1^ j$, $0\leq j\leq k$, form an orthonormal basis, and the sum of their squared norms is

Consequently $\varphi _ k=k^{-1}\log (k+1)$, which tends to zero uniformly.