Proposition 18.26.136457B1027E36

Proof. At $x=[1:0:0]$, put

The homogeneous quotient defines a global nonpositive potential. Its curvature is positive, since $\log (|Z_1|^2+|Z_2|^2)$ is plurisubharmonic in a local trivialization and the remaining background is $(1-a)\omega $. Its only pole is $x$, where it is $a\log |z|^2+O(1)$. By proposition 18.4.8, $p_ x=P_\omega [u_ x]$ is an ideal model with analytic singularities, ordinarily equivalent to $u_ x$. Unitary projective automorphisms preserve $\omega $ and commute with its envelopes for the zero obstacle. Transporting this construction to pairwise distinct points $x_ j$, we obtain models $p_ j$.

Fix $\pi :Y_\pi \to X$ and some Gauduchon form $\gamma _\pi $ on $Y_\pi $. The closed polar current of $p_ j\circ \pi $ is supported on the analytic fiber over $x_ j$. The divisorial restriction in lemma 5.38.1 and the support theorem, [Chapter III, Corollaries 2.11 and 2.14, pp. 141–143, DemCADG], give

where the sum runs over the irreducible curves $E\subset Y_\pi $ with $\pi (E)=x_ j$. Indeed, corollary 18.26.11 shows that the polar current is closed, and after the curve components are removed, the remaining closed current of order zero is supported at finitely many points and therefore vanishes. The sum is finite. Every such curve is contracted by $\pi $, so the restriction of $\pi ^*\omega $ to it is zero. Put

A modification of a surface has finitely many exceptional curves, so there are only finitely many points over which $\pi $ has exceptional curves. Since the points $x_ j$ are distinct, $J_\pi $ is finite. Choose $\varepsilon _\pi >0$ with $\varepsilon _\pi C_{\pi ,j}\leq 1/j$ for every $j\in J_\pi $; if $J_\pi $ is empty, take $\varepsilon _\pi =1$. Now fix

This form is positive and Gauduchon, since on a surface the Gauduchon equation is $\mathrm{dd}^{\mathrm{c}}\eta _\pi =0$. Moreover

These forms are fixed for the whole sequence, and the estimate holds for every modification. Since $\mu _\pi (0)=0$, taking the supremum gives the asserted convergence.

To compute the singularity volumes, let $\rho :Y\to X$ be the blowup at $x$, with exceptional curve $E$, and let $F:Y\to \mathbb P^1$ be the morphism resolving the pencil $[Z_1:Z_2]$. Write $\omega _1$ for the normalized Fubini–Study form on $\mathbb P^1$. Then

For every $v\sim u_ x$, divisorial removal gives $H_ a+\mathrm{dd}^{\mathrm{c}}w\geq 0$, where $w=v\circ \rho -u_ x\circ \rho $ is bounded after extension across $E$. The bounded product of proposition 5.28.2 assigns no mass to $E$, so change of variables off $E$ identifies the non-pluripolar mass of $v$ with $\int _ Y(H_ a+\mathrm{dd}^{\mathrm{c}}w)^2$. Since $H_ a$ is closed, integration by parts for bounded potentials gives

For the last equality, note that $H_ a$ represents the class $H-aE$, where $H=\rho ^*[\omega ]$, and that $H^2=1$, $H\cdot E=0$ and $E^2=-1$. Hence both singularity volumes of $p_ x$, and by unitary transport those of every $p_ j$, are equal to $1-a^2$. The zero model has both volumes equal to one. Since $p_ j\leq 0$, definition 13.22.1 gives $d_\omega (p_ j,0)=1-(1-a^2)=a^2$. This also proves the common positive lower bound for the volumes and the incompatibility of the topologies. $\square$