Theorem 10.28.958E2ED8FCEB9

Proof. Necessity. Let $h$ be a semipositive metric on $L_0$ that extends. By properness we may shrink the disk so that the extending metric is defined on the whole family. On the locus where $f$ is smooth, local integrability and Fubini's theorem show that for almost every $a\in \mathbb C\setminus \{ 0,1\} $ the restriction of the weights of $h$ to $q_ a$ is not identically $-\infty $. Fix such an $a$. On each local product chart of the conic surface of lemma 10.28.8 along $q_ a\setminus \{ p_ a\} $, the weight of the extending metric is not identically $-\infty $, since its central slice is not. By Fubini's theorem, the metric restricts to almost every nearby compact conic, and on such a conic its curvature is a positive measure of degree zero, hence zero. Lemma 10.28.6 therefore makes the central slice of the weight harmonic on each chart; the alternative $-\infty $ is excluded by the choice of $a$.

Consequently, the curvature of $h|_{q_ a}$ is supported at $p_ a$. It is a positive measure of degree one, so it equals $[p_ a]$, which is also the curvature of $h_*|_{q_ a}$. The difference of the two weights is a globally defined harmonic function on the projective line $q_ a$: near $p_ a$, subtracting $\log |w|^2$ from either weight leaves a harmonic function, so the difference extends across $p_ a$. It is therefore constant. Thus the weight difference $u$ between $h$ and $h_*$ is constant along almost every fiber of $f$, away from $\ell $.

Over every point of the base there are local holomorphic sections of $f$ avoiding $\ell $ and the exceptional curves: over the finite chart use $[a:1:1]$, and near infinity use $[1:b:1]$ with $b=1/a$. If $e_ B$ is a local frame of $\mathcal O_{\mathbb P^1}(1)$, then $s_\ell \otimes f^*e_ B$ is a nonvanishing holomorphic frame of $L_0$ along these sections. Composing the weight of $h$ in this frame with a section defines a subharmonic weight on the base. It is not identically $-\infty $, because on almost every fiber the weight is harmonic away from $p_ a$. Two choices of section give the same function at almost every base point, because $u$ is constant on those fibers, and hence everywhere, since subharmonic functions that agree almost everywhere agree everywhere. Changes of the base frame give the transition law of a metric. These weights therefore define a semipositive metric $h_ B$ on $\mathcal O_{\mathbb P^1}(1)$. Fubini's theorem on local product charts, together with the fact that $\ell $ and the exceptional fibers have zero volume, shows that $h$ and $h_\ell \otimes f^*h_ B$ agree almost everywhere on $X_0$. By lemma 5.24.3, their local plurisubharmonic weights agree everywhere, including on the three exceptional fibers. This proves necessity.

Sufficiency. Let $h_ B$ be a semipositive metric on $\mathcal O_{\mathbb P^1}(1)$, and represent it by a homogeneous weight $V$ on $\mathbb C^2\setminus \{ 0\} $, with coordinates $(\xi _1,\xi _2)$ and $a=\xi _1/\xi _2$. In the chart $\xi _2\ne 0$, if $\phi _ B$ is the weight of $h_ B$ in the frame given by the section $\xi _2$, set

The other chart gives the same function, by the transition law of the metric. The function $V$ is plurisubharmonic and satisfies $V(\lambda \xi _1,\lambda \xi _2)=V(\xi _1,\xi _2)+\log |\lambda |^2$ for $\lambda \ne 0$. The weight difference between $h_ B$ and $h_{\mathrm{FS}}$ is bounded above on the compact base, so

Hence $V$ extends plurisubharmonically across the origin, with value $-\infty $ there, by lemma 5.24.7. In each local frame of $L$, the function $V(F_1,F_2)$ is plurisubharmonic, and by homogeneity it transforms as the weight of a metric. Its restriction to each fiber $X_ t$ with $t\ne 0$ is not identically $-\infty $: there the two generating sections define a nonconstant map onto $\mathbb P^1$, and the weight of $h_ B$ is not identically $-\infty $. On the central fiber, the factorization in lemma 10.28.7 and homogeneity give exactly the divisor weight of $\ell $ plus the pullback of the weight of $h_ B$. This is the required extension over $\mathcal X$. $\square$