10.28 Semipositive metrics in a four-point blowup family058F9CA104B3

Proof. Near one of the sections, coordinates centered on it identify the blowup with $\operatorname {Bl}_0(\mathbb C^2)\times \Delta $. Since the four centers are disjoint, these charts prove smoothness and the submersion property. The blowup of the projective morphism $\mathbb P^2\times \Delta \to \Delta $ is projective and proper, and each fiber $X_ t$ is the blowup of $\mathbb P^2$ at the four points $p_ i(t)$.

The quadratic forms

vanish along all four sections and therefore lift to sections of $L$. They are nonzero, which proves that $L$ is effective. For $t\ne 0$, their common zeros in $\mathbb P^2$ are exactly the four points. In suitable affine coordinates $(u,v)$ centered at $p_1,p_2,p_3,p_4$, their linear initial terms are

Each pair is linearly independent. Hence, after the exceptional divisors are subtracted, the transformed sections have no common zero on an exceptional curve. The two transformed sections therefore generate $L_ t$, and $L_ t$ has nonnegative degree on every irreducible curve.

On a fiber, $H^2=1$, $H\cdot E_ i=0$, and $E_ i\cdot E_ j=-\delta _{ij}$. Hence $L_ t^2=0$, and $K_{X_ t}=-3H+\sum _ iE_ i$ gives $K_{X_ t}\cdot L_ t=-2$. A nonzero section of $K_{X_ t}+mL_ t$ would have an effective zero divisor whose intersection with the nef line bundle $L_ t$ is $-2<0$, a contradiction. This argument also excludes a nowhere-vanishing section, whose zero divisor would have intersection zero. $\square$

Proof. Consider the cubic forms

Their common zero set in $\mathbb P^2$ consists of the four central points. In affine coordinates at $p_1$, $p_2$ and $p_3$, their linear initial terms include $(z,y)$, $(z,x)$ and $(z,x-y)$, respectively, while at $p_4$ their quadratic initial terms include $x^2,xy,y^2$. These have no common zero on the corresponding exceptional projective lines. After the simple exceptional factors at $p_1,p_2,p_3$ and the double factor at $p_4$ are removed, the transformed cubics therefore generate $A$.

Their Fubini–Study metric $h_ A$ is smooth and semipositive. Let $h_\ell $ be the divisor metric on $\mathcal O(\ell )$. Since $2L_0=A+\ell $, the square root of $h_ A\otimes h_\ell $ defines a metric $h_0$ on $L_0$, with

The stated form of the weight and of its singularities follows directly. $\square$

Proof. The symmetric matrix of the quadratic form has determinant $-t/2$, so the conic is smooth for $t\ne 0$. It passes simply through all four centers, so its strict transform is a smooth rational curve of class $L_ t$, and its degree against $L_ t$ is $L_ t^2=0$.

At $t=0$ the conic is the union of the lines $\{ z=0\} $ and $\{ x=2y\} $. The second line avoids $p_1,p_2,p_3$ and passes through $p_4$. Each component has multiplicity one at every center it contains, so the central fiber of $\Gamma $ contains no additional exceptional component. The two lines meet at $[2:1:0]$, transversely and away from every center. Since $q=H-E_4$, we have $L_0\cdot q=1$.

Away from the centers, the defining equation at $t=0$ has nonzero derivative in the directions of $\mathbb P^2$ at every point of $q\setminus \{ p\} $, which proves the assertion there. Near $p_4$, use the blowup chart $z=1$, $x=r$, $y=rs$. After the exceptional factor is removed, the equation becomes

At the central point of $q\cap E_4$ we have $r=0$ and $s=1/2$, and the derivative in $s$ is $-2$. Thus $(r,t)$ are local coordinates on $\Gamma $ there. These two cases cover $q\setminus \{ p\} $. $\square$

Proof. Suppose that such a metric exists. Since $\pi $ is proper, we may shrink $\Delta $ so that the whole family lies in its domain: otherwise points outside the neighborhood over $t_ j\to 0$ would have a limit point on $X_0$ outside it. The lower bound makes the restricted weight finite at every point of $q\setminus \{ p\} $, because $\varphi _0$ is smooth there. Hence the restrictions of the weight to $\Gamma $ and to $q$ are not identically $-\infty $.

Near a point of $q\setminus \{ p\} $, choose a product chart $(\zeta ,t)$ on $\Gamma $, as given by lemma 10.28.3, and a holomorphic frame of $L$. The restricted weight $u(\zeta ,t)$ is plurisubharmonic and locally integrable. By Fubini's theorem, for almost every small $t\ne 0$ its restriction to $C_ t$ is not identically $-\infty $, so the metric restricts to $C_ t$. The curvature of this restriction is a positive measure on the compact curve $C_ t$ of total mass $L_ t\cdot C_ t=0$, so it vanishes. Thus $u(\cdot ,t)$ is harmonic for almost every $t$. By lemma 10.28.6 below, whose proof does not use this theorem, the central slice $u(\cdot ,0)$ is harmonic or identically $-\infty $, and it is finite. Hence the restricted weight is harmonic near every point of $q\setminus \{ p\} $.

The restricted metric on $L_0|_ q$ therefore has curvature supported at $p$. This curvature is a positive measure of total mass $L_0\cdot q=1$, so it is the Dirac measure $[p]$. In a coordinate $w$ on $q$ centered at $p$, the restricted weight is therefore $\log |w|^2$ plus a harmonic function, because their difference has zero distributional Laplacian. Since $\ell $ and $q$ meet transversely, $\varphi _0|_ q=\tfrac 12\log |w|^2+O(1)$. The difference $\psi |_ q-\varphi _0|_ q$ thus tends to $-\infty $ at $p$, contradicting the assumed lower bound. $\square$

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. $\square$