Theorem 15.19.7E00C93820852

Proof. Monotonicity of the constants. Let $\alpha \leq \beta $ be Hermitian forms, and let $(u_\alpha ,c_\alpha )$ and $(u_\beta ,c_\beta )$ be the normalized smooth solutions of $(\alpha +\mathrm{dd}^{\mathrm{c}}u_\alpha )^ n=c_\alpha \mu $ and $(\beta +\mathrm{dd}^{\mathrm{c}}u_\beta )^ n=c_\beta \mu $. At a maximum point of $u_\alpha -u_\beta $,

so $c_\alpha \leq c_\beta $. Hence $c_\varepsilon $ decreases as $\varepsilon \downarrow 0$, and lemma 15.19.3 gives $c_0\geq b>0$. Write $c_\eta (\varepsilon )$ for the constant $c_\varepsilon $ defined with the perturbation $\eta $. For two perturbations with $M^{-1}\eta _1\leq \eta _2\leq M\eta _1$, the same comparison gives

Letting $\varepsilon \downarrow 0$ shows that $c_0$ does not depend on $\eta $.

Local estimates. The trace bound of lemma 15.19.4 gives uniform ellipticity on compact subsets of $X\setminus Z$. The smooth coefficients of $\theta +\varepsilon \eta $ and all derivatives of $\log \mu _{\mathrm{loc}}$ have common bounds there. Interior complex Evans–Krylov and Schauder estimates, as in the ellipticity argument of proposition 15.18.5, give uniform $C^ k$ bounds on smaller compact subsets for every $k$.

Convergence and the limiting equation. Normalized compactness in the fixed background $\theta +\eta $, followed by a diagonal extraction over an exhaustion of $X\setminus Z$, gives the stated convergence. The uniform bound $-C\leq u_\varepsilon \leq 0$ passes to the canonical limit, and normalized compactness preserves the supremum zero. Current convergence gives $\theta +\mathrm{dd}^{\mathrm{c}}u_0\geq 0$, and smooth convergence identifies the equation on $X\setminus Z$. The bounded non-pluripolar measure $\mathrm{MA}_\theta (u_0)$ and the smooth measure $\mu $ both vanish on the analytic set $Z$. This proves the equation on all of $X$. For closed $\theta $, bounded closed-background Stokes gives $\int _ X\mathrm{MA}_\theta (u_0)=\int _ X\theta ^ n$, proving the last assertion. $\square$