10.28.5 The image of metric restrictionBBAE9B00E672

Proof. Since $u$ is locally integrable, Fubini's theorem gives $\partial _\zeta \partial _{\bar\zeta }u=0$ in the sense of distributions. On smaller product disks, form the averages

where $\mathrm{d}A$ is planar area. Local upper bounds for $u$ and the reverse Fatou lemma give upper semicontinuity, and Fubini's theorem gives the submean inequality in $\zeta $. The averages are locally integrable, since $u$ is integrable on compact product disks. Thus each $u_\varepsilon $ is subharmonic with zero distributional Laplacian, hence harmonic.

Since $u$ is subharmonic in $t$, the disk averages decrease to $u(\zeta ,0)$ as $\varepsilon \downarrow 0$. A decreasing limit of harmonic functions is harmonic or identically $-\infty $: if the values are bounded below at one point, Harnack's inequality, applied to the differences from the first function, bounds the differences on compact subdisks, and compactness of harmonic functions gives a harmonic limit. This proves the alternative on every smaller disk, and hence on $D_\zeta $. $\square$

Proof. First blow up $p_4$ in $\mathbb P^2$. The map $[x:y]$ then becomes holomorphic: in the chart $x=r$, $y=rs$ with $z=1$, it is $[1:s]$, and the other chart covers the remaining direction. Its fibers are the strict transforms of the lines through $p_4$, and the pullback of $\mathcal O_{\mathbb P^1}(1)$ is $H-E_4$. Next blow up the other three central points, which lie in the directions $a=\infty ,0,1$, respectively. Each affected fiber becomes the union of its strict transform and one exceptional line meeting it, while every other fiber remains a projective line. This proves connectedness of the fibers and the formula for $P$. The equality $L_0=\mathcal O(\ell )+P$ follows from the divisor classes.

In this identification, the restrictions of $F_1,F_2$ to $X_0$ are $s_\ell \otimes f^*x$ and $s_\ell \otimes f^*y$, where $x$ and $y$ are regarded as sections of $\mathcal O_{\mathbb P^1}(1)$. Indeed, at $t=0$ the forms are $xz$ and $yz$; the pullback of $\{ z=0\} $ is $\ell +E_1+E_2+E_3$, and removing the four exceptional factors from the transforms of $xz$ and $yz$ leaves exactly these sections. In a local frame of $L$, the weight $\log (|F_1|^2+|F_2|^2)$ defines a semipositive metric, since a change of frame multiplies both coefficients by the same nonvanishing holomorphic function. On every fiber $X_ t$ with $t\ne 0$ the two sections have no common zero, so the weight is not identically $-\infty $ there. On the central fiber the weight is the sum of the divisor weight of $\ell $ and the pullback of the Fubini–Study weight, which proves the claim and the curvature formula. $\square$

Proof. The symmetric matrix of the quadratic form has determinant $-a(a-1)t/4$, so the conic is smooth for $t\ne 0$; it passes simply through the four centers. Its strict transform has class $L_ t$, so its degree is $L_ t^2=0$. At $t=0$ the conic splits into the two lines $\{ z=0\} $ and $\{ x=ay\} $. Since $a$ is finite and $a\notin \{ 0,1\} $, the line $\{ x=ay\} $ does not pass through the other three central points, so no additional exceptional component occurs, and $q_ a=H-E_4$ has degree one against $L_0$. The two lines meet transversely at $p_ a$, away from all centers.

Away from the centers, the central equation has nonzero derivative in the directions of $\mathbb P^2$ along $q_ a\setminus \{ p_ a\} $. At the point of $q_ a$ on $E_4$, the chart $z=1$, $x=r$, $y=rs$ gives, after the exceptional factor is removed,

At $r=t=0$ and $s=1/a$, the derivative in $s$ is $-a\ne 0$. These charts prove the final assertion. $\square$

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 $\mathcal X$. $\square$

Proof. By theorem 10.28.9, such a metric has the form $h_\ell \otimes f^*h_ B$. Its weight difference from $h_*$ is the pullback of the global quasi-plurisubharmonic difference between the weights of $h_ B$ and $h_{\mathrm{FS}}$, which is bounded above on $\mathbb P^1$. This proves the inequality. That $h_*$ extends is lemma 10.28.7. $\square$