Example 17.3.166EAA0093766C

Example 17.3.16.6EAA0093766C Let $X$ be a compact Kähler curve, so $n=1$, and take $\theta =\omega $ and the reference $V_\omega =0$. For a smooth $\omega $-psh function $u$,

where $\mathrm d^{\mathrm c}$ is the real operator of (5.2.10), so that $\mathrm d\mathrm d^{\mathrm c}=\mathrm{dd}^{\mathrm{c}}$. Indeed, by definition $E_\omega (u)=\frac12\bigl (\int _ Xu\, \omega +\int _ Xu(\omega +\mathrm{dd}^{\mathrm{c}}u)\bigr )$, and Stokes' formula (lemma 13.5.2) gives $\int _ Xu\, \mathrm{dd}^{\mathrm{c}}u=-\int _ X\mathrm du\wedge \mathrm d^{\mathrm c}u$. Since $\mathrm du\wedge \mathrm d^{\mathrm c}u=\frac{\mathrm i}{2\pi }\partial u\wedge \bar\partial u$, the second term is a multiple of the classical Dirichlet energy of $u$.

The unbounded potential $u$ on $\mathbb P^1$ of example 16.22.12, which has full mass, also lies in $\mathcal E^1(\mathbb P^1,\omega )$. Away from its only pole $z=0$ the function $u$ is bounded, and $\mu _ u$ is a finite measure. Near the pole write $r=\log |z|^2$. If $G$ is a smooth function of one real variable, then $\bar\partial G(\log |z|^2)=G'(r)\, \mathrm d\bar z/\bar z$ and $\partial \bar\partial G(\log |z|^2)=G''(r)\, \mathrm dz\wedge \mathrm d\bar z/|z|^2$ on $\mathbb C\setminus \{ 0\} $; since $\mathrm i\, \mathrm dz\wedge \mathrm d\bar z=2\, \mathrm dA$, where $\mathrm dA$ is Lebesgue measure, we get $\mathrm{dd}^{\mathrm{c}}G(\log |z|^2)=\pi ^{-1}G''(r)|z|^{-2}\, \mathrm dA$, which in polar coordinates $z=\rho \mathrm e^{\mathrm i\vartheta }$, with $r=2\log \rho $, is $(2\pi )^{-1}G''(r)\, \mathrm dr\, \mathrm d\vartheta $.

The function $F$ of example 16.22.12 satisfies $F(r)=-\log (-r)$ for $r\leq r_0$, with $r_0$ sufficiently negative, and is therefore smooth there, with $F''(r)=r^{-2}$. On the punctured disc $\{ 0<|z|^2<\mathrm e^{r_0}\} $ the measure $\mu _ u$ is $\mathrm{dd}^{\mathrm{c}}F(\log |z|^2)$, and it does not charge the pole; hence the image of its restriction to $\{ |z|^2<\mathrm e^{r_0}\} $ under $z\mapsto \log |z|^2$ is $F''(r)\, \mathrm dr=r^{-2}\, \mathrm dr$ on $(-\infty ,r_0)$. Since $u-F(\log |z|^2)=-\log (1+|z|^2)$ is bounded near the pole, the finiteness of $\int |u|\, \mu _ u$ near the pole reduces, after the substitution $t=-r$, to

Hence $u$ is a potential of finite energy whose difference from the reference $V_\omega =0$ is unbounded.