Example 16.11.7908479CB91B3

Example 16.11.7.908479CB91B3 On $\mathbb P^1$ let $\theta =\omega _{\mathrm{FS}}=\mathrm{dd}^{\mathrm{c}}\log (1+|z|^2)$ be the Fubini–Study form of example 10.6.7, in the affine coordinate $z$, so that $\int _{\mathbb P^1}\theta =1$. Put

In the coordinate $\zeta =1/z$ near $\infty $ we have $\log (1+|z|^2)=\log (1+|\zeta |^2)-\log |\zeta |^2$, so $\theta =\mathrm{dd}^{\mathrm{c}}\log (1+|\zeta |^2)$ there, while $u_0=-\log (1+|\zeta |^2)$ and $u_\infty =\log (|\zeta |^2/(1+|\zeta |^2))$. Thus the substitution $z\leftrightarrow \zeta $ exchanges the two functions. On $\mathbb C$ we have $\theta +\mathrm{dd}^{\mathrm{c}}u_0=\mathrm{dd}^{\mathrm{c}}\log |z|^2=[0]$ by the Poincaré–Lelong formula (theorem 5.5.6), and $u_0$ is smooth near $\infty $ with $\theta +\mathrm{dd}^{\mathrm{c}}u_0=0$ there. Hence $u_0,u_\infty \in \mathrm{PSH}(\mathbb P^1,\theta )$, with $\theta +\mathrm{dd}^{\mathrm{c}}u_0=[0]$ and $\theta +\mathrm{dd}^{\mathrm{c}}u_\infty =[\infty ]$, the currents of integration at the two points.

Suppose that $w\in \mathrm{PSH}(\mathbb P^1,\theta )$ is a common minorant of $u_0$ and $u_\infty $. Near $0$ the function $u_0$ is $\log |z|^2$ plus a smooth function, so $\nu (u_0,0)=1$ by proposition 6.7.9, and $w\leq u_0$ gives $\nu (w,0)\geq 1$ by the description of Lelong numbers through logarithmic growth in proposition 6.7.1; likewise $\nu (w,\infty )\geq 1$. In dimension one, the Lelong number of a subharmonic function $s$ at a point is the mass that the measure $\mathrm{dd}^{\mathrm{c}}s$ gives to that point (theorem 6.7.6). Hence the positive measure $\theta +\mathrm{dd}^{\mathrm{c}}w$ has atoms of mass at least one at $0$ and at $\infty $, whereas its total mass is $\int _{\mathbb P^1}\theta =1$, because $\langle \mathrm{dd}^{\mathrm{c}}w,1\rangle =\langle w,\mathrm{dd}^{\mathrm{c}}1\rangle =0$. This contradiction shows that $P_\theta (u_0,u_\infty )=-\infty $, although $u_0$ and $u_\infty $ are both $\theta $-plurisubharmonic.