Lemma 16.29.16F6C7CDB99C6E

Proof. Let $s_ D$ be the canonical section of $\mathcal O_ X(D)$, with zero divisor $D$. For a real parameter $t>0$, define a smooth Hermitian metric on this line bundle by prescribing the norm of $s_ D$:

To check that it extends smoothly, use the coordinate $\xi =1/w$ near $D$. The squared norm of a local nonvanishing frame is

which is smooth and positive. The prescribed norm is invariant under the transformations on the complement of $D$, so these local extensions define a global metric. Its curvature form is

Set $\theta =\theta _1$, and let $\sigma =\sigma _0/\pi $ on $E$, where the denominator is the number $\pi $.

We now estimate the curvature from below. Put $r=|v|^2$ and let $H$ be the complex Hessian of $\log (1+tr)$ in the coordinates $(z,w)$. Direct differentiation gives

Consequently

Completing the square gives $\theta _ t\geq -(t/4)\pi ^*\sigma $ on $X\setminus D$, and by continuity this inequality extends across $D$. The restriction of $\theta _ t$ to every fiber is a positive Fubini–Study form, including at infinity. Thus $\theta _ t+A\pi ^*\sigma $ is strictly positive off $D$ whenever $A>t/4$. At $\xi =0$, its coefficient matrix, with the common factor $\mathrm i/(2\pi )$ omitted, is

Its vertical coefficient is positive and its Schur complement is $A-t/4>0$, so the form is strictly positive along $D$ as well. In particular, $\kappa =\theta +\pi ^*\sigma _0$ is Kähler.

The function

extends smoothly across $D$ and satisfies $\theta _ t=\theta +\mathrm{dd}^{\mathrm{c}}f_ t$. Letting $t\downarrow 0$ in the curvature lower bound shows that the class of $\theta $ is nef in the sense of subsection 20.4.16.

Write $\psi _ D=\log |s_ D|_1^2$. By the Poincaré–Lelong formula, $\theta +\mathrm{dd}^{\mathrm{c}}\psi _ D=[D]$. Let $u\in \mathrm{PSH}(X,\theta )$. Then $u-\psi _ D$ is plurisubharmonic on $X\setminus D$. Its lift $\Phi (z,w)$ to $\mathbb C^2$ is invariant under the two deck transformations and, since $u$ is bounded above on the compact surface $X$, satisfies

for some real constant $C$. For each fixed $w$, the function $z\mapsto \Phi (z,w)$ is either identically $-\infty $ or subharmonic. It is $1$-periodic and has logarithmic upper growth, so lemma 5.24.6 makes it constant. Thus $\Phi (z,w)=\phi (w)$, where $\phi $ is a subharmonic function of $w$, not identically $-\infty $. The second deck transformation gives $\phi (w+1)=\phi (w)$, and the bound at $z=0$ gives logarithmic growth, so the same lemma makes $\phi $ constant. Therefore $u=\psi _ D+C$ off $D$, and lemma 5.24.3 extends this equality across $D$. Hence $\theta +\mathrm{dd}^{\mathrm{c}}u=[D]$. $\square$