Lemma 18.47.22BCFCC9C8808

Proof. For $\xi \in \mathbb C^*$, the curve of the pencil

is a smooth rational curve of bidegree $(1,1)$ through both poles, and its $\theta $-degree is two. The restriction of $v$ to this connected curve is either identically $-\infty $ or $\theta |_{C_\xi }$-plurisubharmonic. In the latter case, its curvature measure has total mass two by Stokes' theorem. The local upper bounds of lemma 18.47.1 give Lelong numbers at least one at the two poles, hence, by the identification of Lelong numbers with atoms in lemma 5.37.18, atoms of mass at least one there. These atoms exhaust the measure. The curvature of $\psi |_{C_\xi }$ consists of the same two unit atoms. Thus $v-\psi $ is harmonic, in the sense of distributions, on the compact curve $C_\xi $, and therefore constant.

On the torus use the holomorphic coordinates $(z_1,\xi =z_1z_2)$. Local integrability and Fubini's theorem show that, for almost every $\xi $, the restriction of $v$ to $C_\xi $ is not identically $-\infty $; by the preceding paragraph it is then finite off the poles. Hence the restriction of $v$ to the transverse curve $S=\{ z_1=1\} \simeq \mathbb P^1$ is not identically $-\infty $. Put $u=v|_ S+\log 2$. Since $\theta |_ S=\omega _{\rm FS}$ and $\psi _{\rm FS}|_ S=-\log 2$, the constants on the curves $C_\xi $ give the asserted identity almost everywhere on the torus. Both sides have plurisubharmonic local potentials, so they agree everywhere off the poles. The logarithmic upper bounds give the value $-\infty $ at the poles.

Conversely, $\theta +\mathrm{dd}^{\mathrm{c}}(\psi _{\rm FS}+u\circ f)=f^*(\omega _{\rm FS}+\mathrm{dd}^{\mathrm{c}}u)\geq 0$ off the poles. In a chart at a pole, the local potential is bounded above by $\log (|\zeta _1|^2+|\zeta _2|^2)+C$, since $u$ is bounded above. It therefore extends across the pole as a plurisubharmonic function with value $-\infty $ there, which gives a global $\theta $-plurisubharmonic function dominated by $\psi +C$. $\square$