Example 13.7.6ACAEA8D8BE87

Example 13.7.6.ACAEA8D8BE87 On $\mathbb P^1$ let $\omega _{\mathrm{FS}}$ be the Fubini–Study form of example 10.6.7, which has total mass one, and put $u=\log |z|^2-\log (1+|z|^2)$ and $u_ k=\max \{ u,-k\} $. As computed in that example, $u$ extends smoothly across infinity and $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}u=[0]$. Each $u_ k$ is bounded and $\omega _{\mathrm{FS}}$-plurisubharmonic, so $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}u_ k$ has mass one by proposition 13.5.1. The sequence $u_ k$ decreases to $u$, hence converges to it in extended capacity (lemma 13.7.3), and the measures $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}u_ k$ converge weakly to $[0]$, because $u_ k\to u$ in $L^1$. However, $\langle [0]\rangle =0$ by example 13.5.6. The weak limit retains the mass that the non-pluripolar product discards at the pole, so the mass hypothesis in theorem 13.7.5 cannot be omitted.