Theorem 15.19.17E621E67CCA9A

Proof. Existence follows from corollary 15.19.12. For uniqueness we use an energy identity, which is available because $\theta $ is closed. Fix one cluster limit $u_0$ and put $A=\theta +\mathrm{dd}^{\mathrm{c}}u_0$. This is a positive closed current with bounded local potentials, smooth and strictly positive on $X\setminus E$ by the trace bounds. If $v$ is another bounded solution with scalar $c$, bounded closed-background Stokes gives $c=\int _ X\theta ^3=c_0$. Put $B=\theta +\mathrm{dd}^{\mathrm{c}}v$ and $w=v-u_0$. The bounded integration-by-parts calculus of section 5.27 gives

All three energies are nonnegative, so each of them vanishes, in particular the one with $A^2$. On every relatively compact chart of $X\setminus E$, the smooth positive factor $A^2$ controls the weak gradient of $w$, so $\mathrm{d}w=0$ there. The complement of a proper complex analytic subset in the connected manifold $X$ is connected. Thus $w$ is constant almost everywhere off $E$. Since $E$ has ordinary volume zero, canonical almost-everywhere equality of quasi-plurisubharmonic functions, lemma 5.24.3, gives $v=u_0+C$ on all of $X$. Normalization fixes $C=0$.

The compactness and smooth local estimates of theorem 15.19.7 now make every subsequential limit equal to $u_0$, proving convergence of the whole family. The equation and its unique normalized solution do not depend on $\eta $. Descent and the zero exceptional mass identify bounded solutions upstairs and downstairs, proving the final assertion. $\square$