Example 16.22.12246496245694

Example 16.22.12.246496245694 On $\mathbb P^1$ let $\omega =\mathrm{dd}^{\mathrm{c}}\log (1+|z|^2)$ in the affine coordinate $z$, so that $\int _{\mathbb P^1}\omega =1$; since $\omega $ is Kähler, $V_\omega =0$. Define

The function $F$ is continuous and nondecreasing. Since $-\log (-r)<r+2$ at $r=-1$, there is $\delta >0$ with $F(r)=r+2$ for $r\geq -1-\delta $, and since $r+2+\log (-r)\to -\infty $ as $r\to -\infty $, we have $F(r)=-\log (-r)$ for all sufficiently negative $r$.

We first show that $F(\log |z|^2)$ is subharmonic on $\mathbb C\setminus \{ 0\} $. On the open set $\{ |z|^2>\mathrm e^{-1-\delta }\} $ it equals the harmonic function $\log |z|^2+2$. On the open set $\{ 0<|z|^2<\mathrm e^{-1}\} $ it equals $\max \{ g(\log |z|^2),\log |z|^2+2\} $, where $g(t)=-\log (-t)$ for $t<0$. Since $g''(t)=t^{-2}>0$, the function $g$ is convex, so it is the supremum of its tangent lines $t\mapsto g(s)+g'(s)(t-s)$, $s<0$, whose slopes $g'(s)=-1/s$ are positive. Hence $g(\log |z|^2)$ is a continuous supremum of the harmonic functions $g(s)+g'(s)(\log |z|^2-s)$; it therefore satisfies the submean inequality and is subharmonic, and so is its maximum with $\log |z|^2+2$. Since subharmonicity is a local property, $F(\log |z|^2)$ is subharmonic on $\mathbb C\setminus \{ 0\} $; it tends to $-\infty $ at $0$ and extends subharmonically across the polar set $\{ 0\} $ by theorem 3.4.27.

Next we check that $u$ is $\omega $-plurisubharmonic. For $|z|^2\geq \mathrm e^{-1}$ we have $u=2-\log (1+|z|^{-2})$. In the coordinate $\zeta =1/z$ near $\infty $ this reads $u=2-\log (1+|\zeta |^2)$, while $\omega =\mathrm{dd}^{\mathrm{c}}\log (1+|\zeta |^2)$, because $\log (1+|z|^2)=\log (1+|\zeta |^2)-\log |\zeta |^2$ and $\log |\zeta |^2$ is pluriharmonic for $\zeta \ne 0$; thus $u$ is smooth near $\infty $, with $u(\infty )=2$ and $\omega +\mathrm{dd}^{\mathrm{c}}u=0$ there. Hence $u\in \mathrm{PSH}(\mathbb P^1,\omega )$, with $\omega +\mathrm{dd}^{\mathrm{c}}u=\mathrm{dd}^{\mathrm{c}}F(\log |z|^2)$ on $\mathbb C$, and its only pole is $0$.

To compute the mass of $u$, note that its Lelong number at $0$ is that of the subharmonic function $F(\log |z|^2)$, which differs from $u$ by a smooth function, and since $F$ is continuous and nondecreasing, $\sup _{|z|<\mathrm e^{t/2}}F(\log |z|^2)=F(t)$. By proposition 6.7.1, the Lelong number is therefore

By theorem 6.7.6, in one variable the mass that $\mathrm{dd}^{\mathrm{c}}F(\log |z|^2)$ gives to $0$ is this Lelong number, namely the limit of the masses of the discs $B(0,r)$ as $r\downarrow 0$, so the current $\omega +\mathrm{dd}^{\mathrm{c}}u$ has no atom at its pole. By proposition 13.4.7, $\mu _ u$ is the restriction of $\omega +\mathrm{dd}^{\mathrm{c}}u$ to $\{ u>-\infty \} $, hence the mass of $u$ is $\int _{\mathbb P^1}\langle \omega +\mathrm{dd}^{\mathrm{c}}u\rangle =\int _{\mathbb P^1}\omega =1$.

The potential $0$ is nonpositive, less singular than $u$, and has the same mass $\int _{\mathbb P^1}\omega =1$, so theorem 16.22.11 gives $P_\omega [u]\geq 0$; since also $P_\omega [u]\leq V_\omega =0$, we obtain $P_\omega [u]=0$. Near $0$, however, $u=-\log (-\log |z|^2)+O(1)$ is unbounded below. Thus the singularity envelope can change the singularity type of a potential while preserving its full mass.