Theorem 10.24.78D207AFD99F5

Proof of theorem 10.24.7. The background and the pole. Use the incidence surface of theorem 10.24.1, with blowup map $\pi :Y\to \mathbb P^2$ and exceptional curve $E$ over $[1:0:0]$, and let $\beta ,\gamma $ be the pulled-back Fubini–Study forms as there. Put $D=\pi ^{-1}(Z_0=0)$ and $A=\pi ^*c_1(\mathcal O(1))$. Then $E\cap D=\varnothing $, $A^2=1$ and $A\cdot E=0$. The class $A$ is nef, since it has the smooth semipositive representative $\beta $. Moreover $\{ \gamma \} =A-[E]$, and $\beta +\varepsilon \gamma $ is Kähler for $\varepsilon >0$. Thus

is a Kähler current in $A$, which proves bigness.

Choose an even nonnegative smooth function $\zeta $, supported in $[-1,1]$, positive on its interior and of integral one. The smooth convex function

satisfies $U=0$ for $t\leq -1$, $U=t$ for $t\geq 1$, and $U',U''\geq 0$. On $Z_0\ne 0$, write $z=(Z_1/Z_0,Z_2/Z_0)$, $s=|z|^2$ and $t=\log s$, and give the section $Z_0$ of $\mathcal O(1)$ the squared norm $\mathrm{e}^{-U(t)}$; this norm is constant near the affine origin. Near the line at infinity, use

The section $Z_1$ then has squared norm $\mathrm{e}^{-U}/|w|^2=1/h$ near $w=0$. These formulas define a smooth Hermitian metric on $\mathcal O(1)$. Pull it back to $\mathcal O_ Y(D)$ and let $\sigma =\pi ^*Z_0$. Its curvature form $\theta $ and the norm of $\sigma $ are

The radial Hessian shows that $\theta $ is semipositive. It vanishes near $E$ and has rank one near $D$. In particular $V_\theta =0$, and the nef class has volume $A^2=1$.

Extension of the potential. On $Y\setminus (E\cup D)\simeq \mathbb C^2\setminus \{ 0\} $, set

so that $\theta +\mathrm{dd}^{\mathrm{c}}\rho =\Omega $. The eigenvalues of the Hermitian matrix of $\Omega $ are $p/s$ and $p'/s$, which are positive. Extend $\rho $ to $E$ by $B$ and to $D$ by $-\infty $.

Near the affine origin, the smooth functions $F(\log (|z|^2+\varepsilon ))$ are plurisubharmonic and decrease to the continuous extension of $F(\log |z|^2)$ as $\varepsilon \downarrow 0$. Indeed, their two Hessian eigenvalues are $p/(s+\varepsilon )$ and $(s p'+\varepsilon p)/(s+\varepsilon )^2$, with $p,p'$ evaluated at $\log (s+\varepsilon )$. Since $\theta =0$ there, pulling back to $Y$ proves positivity near $E$.

Near $D$ put $t_\varepsilon =\log h-\log (|w|^2+\varepsilon )$ and $g(t)=F(t)-t$. The local potentials $g(t_\varepsilon )+\log h$ decrease to $\rho +\log h$, and their Hessians are

Thus $\rho $ is $\theta $-plurisubharmonic on $Y$. Since $\rho $ is bounded near $E$ and grows like a double logarithm near $D$, all its Lelong numbers vanish.

The volume form and full mass. With Euclidean area $\mathrm{d}A$ in each complex variable, the radial determinant is

Define $\mathrm{d}V_ Y=\mathrm{e}^{-F}\Omega ^2$ off the two divisors. In an exceptional chart $z=(u,uv)$, the real Jacobian is $|u|^2$ and $s=|u|^2h$, so the left profile identity gives

At infinity, the map $z=(1/w,v/w)$ has real Jacobian $|w|^{-6}$, and the right profile identity gives

The analogous second charts cover each divisor. The extensions are smooth and strictly positive, and they agree on overlaps because they agree on the punctured overlaps. The last expression exhibits the nonintegrable pole with smooth positive coefficient.

On the complement of the two divisors, $\mathrm{e}^{\rho }|\sigma |^{-2}\mathrm{d}V_ Y=\Omega ^2$; we extend this density by zero on $D$. Both measures give zero mass to the two divisors. In polar coordinates on $\mathbb C^2$, the Euclidean volume is $\pi ^2s\, \mathrm{d}s$ times the normalized angular measure, so

This proves the global equation and shows that $\rho $ has full mass.

Bounded truncations and energy. Near $D$ write $\alpha =\mathrm{dd}^{\mathrm{c}}t=\mathrm{dd}^{\mathrm{c}}\log h$ and $\beta _ t=(\mathrm{i}/2\pi )\partial t\wedge \bar\partial t$. These forms have rank one, and

Integration of $\alpha \wedge \beta _ t$ over the angular and projective variables gives $\mathrm{d}t$. Since $|\rho |=O(\log t)$ and $p'=O(t^{-2})$, the integrals of $|\rho |$ against $\Omega ^2$ and $\Omega \wedge \theta $ near $D$ are finite. Near $E$ the potential is bounded and $\theta =0$, and the remaining region is compact with smooth data.

Choose $M$ so that $\phi =\rho -M\leq 0$, and set $\phi _ k=\max (\phi ,-k)$. For large $k$ the truncation changes $\phi $ only in the cusp near $D$, where $\phi =\log H(p(t))-M$ is strictly decreasing. If $\phi (T_ k)=-k$, then $k=2\log T_ k+O(1)$. On $\mathbb C^2\setminus \{ 0\} $, the radial local potential of $\theta +\mathrm{dd}^{\mathrm{c}}\phi _ k$ is

Near $D$, in the frame used above, the local potential of $\theta +\mathrm{dd}^{\mathrm{c}}\phi _ k$ is $G_ k(t)+\log |w|^2$, which equals the smooth function $\log h-k$; the term $\log |w|^2$ cancels the divisorial part of $\mathrm{dd}^{\mathrm{c}}G_ k(t)$. At $t=T_ k$, the derivative of $G_ k$ jumps upward from $p(T_ k)$ to $1$.

Convolving $G_ k$ in $t$ with an even nonnegative smooth kernel gives convex functions decreasing to $G_ k$ as the scale decreases. The corresponding bounded local potentials converge uniformly on compact sets, so their Bedford–Taylor products converge. Integrating the smooth radial measures through the transition shows that the transition carries mass $1-p(T_ k)^2$ for $(\theta +\mathrm{dd}^{\mathrm{c}}\phi _ k)^2$ and mass $1-p(T_ k)$ for $(\theta +\mathrm{dd}^{\mathrm{c}}\phi _ k)\wedge \theta $. Beyond the transition, $G_ k$ is linear and both top-degree measures vanish. Since $\phi _ k=-k$ on the transition, its energy contributions have absolute values $k(1-p(T_ k)^2)$ and $k(1-p(T_ k))$, and both tend to zero because $1-p(T_ k)\sim 2/T_ k$. The contributions of the remaining region converge absolutely by the bounds above. At $E$, bounded potentials give no mass to the curve, in accordance with the vanishing radial boundary mass $p(-\infty )^2=0$. Consequently

has a finite limit, and in particular is bounded below. This proves $\rho \in \mathcal E^1(Y,\theta )$.

Uniqueness follows from the full-mass comparison argument in the proof of theorem 10.24.1, now with $\mathrm{d}\mu =|\sigma |^{-2}\mathrm{d}V_ Y$ off $D$, extended by zero on $D$. Its total mass is infinite, but each exponential density $\mathrm{e}^{\rho }\, \mathrm{d}\mu $ and $\mathrm{e}^{u}\, \mathrm{d}\mu $ is a finite measure of mass $1$. Comparison on the two strict sublevel sets forces equality almost everywhere, and lemma 5.24.3 gives equality of the potentials everywhere.

The two metric ends. Fix $v=v_0$ in an exceptional chart and write $h_0=1+|v_0|^2$. The restriction of $\Omega $ to the normal line $\{ v=v_0\} $ is

Its coefficient is asymptotic to $(3c\sqrt{h_0}/4)|u|^{-1}$, so the current has no smooth extension at any point of $E$. The radial path $u=r>0$ has finite remaining length, bounded by $C\int _0^{r_0}r^{-1/2}\, \mathrm{d}r$. At $D$ the normal coefficient is asymptotic to $2/(|w|^2t^2)$ and also diverges. Thus every open set on which $\Omega $ is smooth and strictly positive misses both curves.

Let $W$ be a dense Zariski open set on which $\Omega $ is smooth and strictly positive. Its complement $Y\setminus W$ is a proper analytic set containing $E$, and by local finiteness of irreducible components and compactness it has finitely many components, which are curves or points. Each curve other than $E$ meets $E$ in finitely many points. Choose a point of $E$ that avoids all these intersections and all point components. In a sufficiently small chart around it, the complement of $W$ is exactly $E$, so a tail of the radial path above lies in $W$. Its points approaching $E$ form a Cauchy sequence for the intrinsic metric with no limit in $W$: the smooth positive metric induces the manifold topology of $W$, and the only ambient limit is the removed point. Hence $W$ is incomplete.

Finally, on the cusp $\Omega =p\mathrm{dd}^{\mathrm{c}}t+p'\, \mathrm{d}t\wedge \mathrm d^{\mathrm c}t$, both terms are nonnegative, and $p'\geq c_0/t^2$ for large $t$ and some $c_0>0$. The length of any path with $t\to \infty $ is therefore bounded below by a positive multiple of $\int |\mathrm{d}t|/t$, which diverges. Thus the end at finite distance is the null curve $E$, disjoint from the pole. $\square$