Lemma 23.1.731C279783821E

Proof. Start with smooth $f_ j=t_ ju_ j$ realizing the shrinking defect $(m/6)\omega ^3=\mathcal L(m\, \omega _{\mathrm{FS}}^2)$, and put $H_ j=\mathrm{dd}^{\mathrm{c}}f_ j+t_ j\omega $. By lemma 23.1.72, $H_ j^2\rightharpoonup (m/6)\alpha ^2$, while $H_ j$ has trace mass $O(t_ j)$. Fix $A\in GL(3,\mathbb C)$, regarded as an automorphism of $X$, and let $a$ be the induced projective transformation of $\mathbb P^2$. Choose $C_ A\geq 1$ with $A^*\omega \leq C_ A\omega $, and put

For this fixed $A$, the trace mass of $A^*H_ j$ is $O_ A(t_ j)$, and the positive difference $(H_ j^ A)^2-A^*(H_ j^2)$ has trace mass $O_ A(t_ j^2)$. Since $\alpha =\pi ^*\omega _{\mathrm{FS}}$, it follows that

The limiting measure has mass $m$, so the integral identity of lemma 23.1.68, applied with the parameter $C_ At_ j$, gives $d_ j^ A=M_\omega (f_ j\circ A)-6\to m$. For large $j$, multiply $f_ j\circ A$ by $\sqrt{m/d_ j^ A}\to 1$. The mass is then exactly $6+m$, and the weak limit is unchanged. The underlying normalized potential is $u_ j\circ A/C_ A\in \mathrm{PSH}(X,\omega )$, with parameter $C_ At_ j\sqrt{m/d_ j^ A}\to 0$. The total-variation estimate of lemma 23.1.68 therefore shows that $\mathcal L(m\, a^*\omega _{\mathrm{FS}}^2)$ is a shrinking defect; any prescribed bounds $\varepsilon _ j$ are met by passing to a sufficiently late subsequence.

Now take $A_ l=\operatorname {diag}(R_ l^{-1},1,1)$, $R_ l\to +\infty $. Since

and $a_ l^{-1}[q_0:q_1:q_2]=[R_ lq_0:q_1:q_2]\to [1:0:0]$ outside the hyperplane $\{ q_0=0\} $, which has zero volume, dominated convergence gives $a_ l^*\omega _{\mathrm{FS}}^2\rightharpoonup \delta _{[1:0:0]}$. By weak continuity of $\mathcal L$, the shrinking defects $\mathcal L(m\, a_ l^*\omega _{\mathrm{FS}}^2)$ converge to $mh_{[1:0:0]}$. For each $l$, choose one corrected potential for $A_ l$, sufficiently late in its sequence that its parameter is less than $\varepsilon _ l$ and its measure differs from the corresponding limit by less than $1/l$ on the first $l$ functions of a fixed dense sequence of test functions. This diagonal sequence has the exact mass $6+m$ and the limit $\omega ^3+mh_{[1:0:0]}$. Unitary automorphisms move this fiber to any prescribed fiber. $\square$