Proposition 14.19.44877A402699F6

Proof. Put $\theta =\chi +\mathrm{dd}^{\mathrm{c}}\underline u$ and $w=u-\underline u-\sup _ X(u-\underline u)\leq 0$. Since $\Gamma \subseteq \Gamma _1$, we have $\operatorname {tr}_\omega \mathrm{dd}^{\mathrm{c}}w\geq -\operatorname {tr}_\omega \theta \geq -C$. The Green-kernel argument in the proof of proposition 14.11.1, applied to the operator $\operatorname {tr}_\omega \mathrm{dd}^{\mathrm{c}}$, therefore bounds $\| w\| _{L^1}$ uniformly.

Center a holomorphic coordinate ball of fixed size at a minimum point of $w$, and rescale it to the real unit ball. For a fixed sufficiently small $\varepsilon >0$, set $q=w+\varepsilon |x|^2$ and $m=\min _ Xw$. The real contact-set construction of (14.11.2) gives a compact set $E$ on which

On $E$, $\mathrm{dd}^{\mathrm{c}}w\geq -C\varepsilon \omega $. Choose $\varepsilon $ so that $C\varepsilon <r$. Then $P=\omega ^{-1}(\mathrm{dd}^{\mathrm{c}}w+r\omega )\geq 0$ and

Lemma 14.19.43 bounds $P$, and hence the complex trace of the Hessian of $q$. The nonnegative real Hessian of $q$ on $E$ therefore has bounded trace, and hence bounded determinant, so the contact inequality implies $|E|\geq c>0$. Since $w\leq m+\varepsilon /2$ on $E$, the $L^1$ bound gives $c(-m-\varepsilon /2)\leq C$. This bounds the oscillation of $w$, hence that of $u$; with the normalization $\inf _ Xu=0$, this is the assertion. $\square$