Proposition 12.6.47C7D8502937F2

Proof. Write $v_ j=c_ j/e_ j$ and $\psi _ j=\phi _ j/b_ j$. On $\mathscr S$ the equation for $\phi _ j$ on $B$ becomes

At the end where $\theta _0=\theta _+(b)$, $v_ j\to \infty $. Set $\epsilon =1/v$ and $H_\epsilon =\int _ B\sigma /(a+\epsilon )$. Then $F_ v=[H_\epsilon (a+\epsilon )]^{-1}$ and $H_\epsilon \to H_ a$. At the end where $\theta _0=\theta _-(b)$, $v_ j\to -1/M$. With $f=1-a/M$ and $t=v+1/M>0$, the denominator of $F_ v$ is $Q(v)(f+ta)$, and $Q(v)\to H_-$. Thus the respective limits of $F_{v_ j}$ are

Choose $1<p<r$, where $r>1$ is the ratio $r_0$ or $r_ M$ of that end. Integration over tubes as in the proof of proposition 12.6.44 gives $a^{-1}\in L^ p(B,\sigma )$, respectively $f^{-1}\in L^ p(B,\sigma )$. Near the first end $F_ v\leq C/a$, and near the second $F_ v\leq C/f$. Since the extreme sets have volume zero, dominated convergence gives $F_{v_ j}\to F_\pm $ in $L^ p(B,\sigma )$, and each limiting density has integral one. Since $\sigma /\kappa ^ m$ is smooth and strictly positive on compact $B$, the densities of the equations also have a common bound in $L^ p(B,\kappa ^ m)$.

We apply theorem 14.19.21 to the fixed background $\kappa $: every subsequence of $(\psi _ j)$ has a uniformly convergent further subsequence. Continuity of bounded products under uniform convergence identifies its limit with a normalized solution of $(\kappa +\mathrm{dd}^{\mathrm{c}}\psi _0)^ m=F_\pm \sigma $. Since $\kappa $ is closed, it satisfies the Guan–Li condition, so 14.25.2 makes this solution unique. Every cluster limit is consequently the same, which proves uniform convergence of the whole sequence. Since $b_ j\to b>0$, we obtain $u_ j\to u_0=(b\psi _0)\circ p_ B$ uniformly.

To prove convergence of the products, note that on each coordinate ball the convergence of the background coefficients provides a common smooth function $q$, a quadratic polynomial, such that all $u_ j+q$ are plurisubharmonic. These corrected potentials are uniformly bounded and converge uniformly. The continuity of bounded products under convergence in capacity (section 5.27) gives weak convergence of all their products, with uniform local mass bounds. Expanding the products as in (5.28.1), and using smooth convergence of the remaining background factors, we obtain the asserted convergence of currents. In top degree, $(\theta _ j+\mathrm{dd}^{\mathrm{c}}u_ j)^ n=\mu $ for all $j$, so the limit solves the endpoint equation.

For uniqueness, let $v=\chi \circ p_ B$ be a bounded solution pulled back from $B$, with $\sup _ B\chi =0$. The current $A_\chi =b\kappa +\mathrm{dd}^{\mathrm{c}}\chi $ on $B$ is positive, as one sees by pairing the pulled-back current $\theta _0+\mathrm{dd}^{\mathrm{c}}v$ with $w$ and with pullbacks of positive forms on $B$. The product expansion gives $nL_0A_\chi ^ m\wedge w=\sigma \wedge w$, which determines $A_\chi ^ m=\sigma /(nL_0)$ off the extreme set. That density already has total mass $H_0/n=b^ m$, whereas mass invariance for the closed background $b\kappa $ gives $\int _ BA_\chi ^ m=b^ m$. Positivity therefore excludes any mass of $A_\chi ^ m$ on the extreme set. Thus $\chi /b$ satisfies the same normalized equation on $B$ as $\psi _0$, and 14.25.2 proves the asserted uniqueness. $\square$