PomeExample 13.28.10.98D940133E3B On $\mathbb P^1$ with the Fubini–Study form $\theta $ of area one, there are zero-mass ideal models $p_ j,p$ with $p_ j\to p$ in capacity but $d_\theta (p_ j,p)=2$ for every $j$.
Example 13.28.1098D940133E3B
Proof. In the affine coordinate $z$, where $\theta =\mathrm{dd}^{\mathrm{c}}\log (1+|z|^2)$, put for $a\in \mathbb C$
\[ p_ a(z)=\log \frac{|z-a|^2}{(1+|z|^2)(1+|a|^2)}. \]
This function extends to $\mathbb P^1$, has supremum zero by the Cauchy–Schwarz inequality, and satisfies $T_{p_ a}=\delta _ a$. Hence it is the ideal model with polar measure $\delta _ a$, by lemma 20.3.20. Choose $a_ j\ne 0$ tending to zero and set $p_ j=p_{a_ j}$, $p=p_0$. On the complement of any fixed disk about zero the convergence is uniform, including near infinity. Small disks about zero lie in sublevel sets $\{ p_0<-t\} $ with $t$ large, so their capacities tend to zero as the radius tends to zero, by (5.29.4). Therefore $p_ j\to p$ in capacity. Since $\| \delta _{a_ j}-\delta _0\| _{\mathrm{TV}}=2$, (13.28.5) gives the asserted distance. $\square$
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