Example 13.15.17446B53338035

Example 13.15.17.446B53338035 On $X=\mathbb P^1$, let $z$ be the standard affine coordinate and let $\delta _0$ denote the unit Dirac measure at its origin. Take the Fubini–Study form

and the global nonpositive potential

At infinity, in the coordinate $\zeta =1/z$, the function $u$ equals $-\log (1+|\zeta |^2)$, so it is smooth there. The Poincaré–Lelong formula gives

The second identity holds because the extended measure discards the pole set and the current vanishes on its complement.

Each $u_ j$ is bounded and $\omega $-plurisubharmonic. Its measure $\mu _ j=\mu _{u_ j}=\omega +\mathrm{dd}^{\mathrm{c}}u_ j$ has mass one by Stokes' theorem, and $\mu _ j=0$ on the open set $\{ u>-j\} $ by locality. Hence

These disks shrink to the origin, so $\mu _ j\rightharpoonup \delta _0$. On their supports, $u_ j=-j$. Consequently, for $j>T$,

Nevertheless, $u_ j\to u$ in capacity. Indeed, outside any neighborhood of the origin the functions eventually agree with $u$, and the origin has arbitrarily small neighborhoods of arbitrarily small capacity. Thus bounded mass and capacity convergence do not prevent the undamped measures from acquiring mass at a pole of the limit. The damped limit is the extended measure of $u$, as predicted by theorem 13.15.9.