Proposition 13.4.10.FB5ACFB46DB8 Let $\mu $ be a probability measure supported on a compact set $K\subset \mathbb C\subset \mathbb P^1$, and let $\omega _{\mathrm{FS}}$ be the Fubini–Study form of example 10.6.7, which has total mass one on $\mathbb P^1$. Then
\[ q(z)=\int _ K\log \frac{|z-a|^2}{(1+|z|^2)(1+|a|^2)}\, \mathrm d\mu (a) \]
defines a nonpositive $\omega _{\mathrm{FS}}$-plurisubharmonic function on $\mathbb P^1$ with $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q=\mu $. Its Lelong number at every point $x\in \mathbb P^1$ is $\mu (\{ x\} )$, where $\mu $ is regarded as a measure on $\mathbb P^1$.
Proof.
In the affine chart put $\rho (z)=\log (1+|z|^2)$, so that $\omega _{\mathrm{FS}}=\mathrm{dd}^{\mathrm{c}}\rho $. The integrand is nonpositive, since $|z-a|^2\leq (|z|+|a|)^2\leq (1+|z|^2)(1+|a|^2)$ by the Cauchy–Schwarz inequality. On the affine chart,
\[ q+\rho =p_\mu -c,\qquad p_\mu (z)=\int _ K\log |z-a|^2\, \mathrm d\mu (a),\qquad c=\int _ K\log (1+|a|^2)\, \mathrm d\mu (a). \]
Since $\log |z-a|^2=4\pi E_2(z-a)$ with the fundamental solution $E_2$ of (3.1.14), the function $p_\mu =4\pi E_2*\mu $ is subharmonic and locally integrable with $\Delta p_\mu =4\pi \mu $, by proposition 3.4.11. Since $\mathrm{dd}^{\mathrm{c}}=\Delta /(4\pi )$ on functions of one complex variable, as planar measures, this gives $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q=\mathrm{dd}^{\mathrm{c}}p_\mu =\mu $ on the affine chart, and $q=p_\mu -c-\rho $ is quasi-plurisubharmonic there.
Near infinity use the coordinate $w=1/z$, in which the integrand becomes
\[ \log |1-aw|^2-\log (1+|w|^2)-\log (1+|a|^2). \]
Let $R_ K=\max _{a\in K}|a|$. For $|w|<1/(2R_ K+2)$ we have $|1-aw|\geq 1/2$, so this expression is a smooth function of $(a,w)$ on $K\times \{ |w|<1/(2R_ K+2)\} $ whose derivatives in $w$ of every order are bounded uniformly in $a\in K$; at $w=0$ it takes the value $-\log (1+|a|^2)$. Differentiating under the integral sign, we see that $q$ extends to a smooth function near infinity, and that $q+\log (1+|w|^2)=\int _ K\log |1-aw|^2\, \mathrm d\mu (a)-c$ is harmonic there, because each $\log |1-aw|^2$ is harmonic in $w$. Since $\omega _{\mathrm{FS}}=\mathrm{dd}^{\mathrm{c}}\log (1+|w|^2)$ in this chart and $\mu $ vanishes near infinity, $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q=0=\mu $ there. Hence $q$ is quasi-plurisubharmonic on $\mathbb P^1$, $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q=\mu \geq 0$ globally, and $q$ is a nonpositive $\omega _{\mathrm{FS}}$-plurisubharmonic function.
At infinity both the Lelong number and the point mass vanish, because $q$ is smooth there and $K\subset \mathbb C$. At an affine point $x$, the Lelong number of $q$ is that of the local subharmonic function $q+\rho $ (corollary 6.9.3), and by (6.7.8) and (6.7.5) it equals $\lim _{r\downarrow 0}\mu (B(x,r))=\mu (\{ x\} )$.
$\square$
Pome
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