Proposition 14.24.137F590FD94846

Proof. Choose finitely many coordinate balls $B_ j=\{ |z_ j|<R_ j\} $, with closures in their charts, such that the concentric smaller balls $\{ |z_ j|<r_ j\} $ cover $X$. On $B_ j$, let $\psi _ j$ solve the Dirichlet problem

The density of $g\omega ^ n$ with respect to coordinate volume is smooth and strictly positive. The classical smooth Dirichlet theorem therefore applies, since $C_ j(|z_ j|^2-R_ j^2)$ is a smooth strict subsolution with zero boundary values for sufficiently large $C_ j>0$. Thus $\psi _ j$ is smooth and strictly plurisubharmonic on $\overline B_ j$, and the maximum principle gives $\psi _ j\leq 0$.

We need a lower bound for $\psi _ j$ independent of $\sup _ Xg$. Write the coordinate equation as $\det (\partial _ a\bar\partial _ b\psi _ j)=D_ j$, and put $M_ j=-\min _{B_ j}\psi _ j$. Let $x_ j$ be a minimum point. For each real slope $p$ with $|p|<M_ j/(2R_ j)$, the function $\psi _ j-p\cdot (x-x_ j)$ attains its minimum at an interior point, because its value at $x_ j$ is $-M_ j$, while its boundary values are larger than $-M_ j$. At such a contact point the real Hessian $S$ of $\psi _ j$ is nonnegative, and the gradient of $\psi _ j$ maps the contact set onto the ball of these slopes. The area formula and the matrix inequality

therefore give $M_ j\leq C(n,R_ j)\| D_ j\| _{L^2(B_ j)}^{1/n}$. To prove the matrix inequality, let $J$ be the standard complex structure on the real coordinate space and average $S$ with $J^\mathsf TSJ$. Concavity of $\log \det $, and the fact that the eigenvalues of this $J$-invariant average occur in pairs, give the displayed factor; singular matrices are handled by approximation. Since the ratio of $\omega ^ n$ to coordinate volume is fixed and bounded, $\| D_ j\| _2$ is controlled by $B_0$. Consequently

where $M_0$ depends only on $X,\omega ,B_0$ and the chosen finite cover.

Next we glue the local solutions. Put $\beta _ j=\mathrm{dd}^{\mathrm{c}}|z_ j|^2$. Choose $a_ j>0$ such that $a_ j\beta _ j\leq \omega /2$ on $B_ j$, and let $t_ j=(R_ j^2+r_ j^2)/2$. Fix $\tau >0$ so small that $\tau M_0<a_ j(R_ j^2-r_ j^2)/4$ for every $j$. The smooth function

is bounded below by a fixed positive constant on the smaller ball, negative on a collar of $\partial B_ j$, and satisfies $\omega _{s_ j}\geq \omega /2+\tau \mathrm{dd}^{\mathrm{c}}\psi _ j$. Choose a smooth convex function $\phi _ j:\mathbb R\to \mathbb R$ approximating $s\mapsto \max (s,0)$, equal to zero below a small negative threshold and to $s$ above the corresponding positive threshold, with $0\leq \phi _ j'\leq 1$ and $\phi _ j''\geq 0$; the thresholds are chosen smaller in absolute value than both margins of $s_ j$. Set $Q_ j=\phi _ j(s_ j)$ on $B_ j$ and extend it by zero. This extension is smooth, and

On the smaller ball, $Q_ j=s_ j$ and $\omega _{Q_ j}\geq \tau \mathrm{dd}^{\mathrm{c}}\psi _ j$. If $N$ is the number of balls, put $q=N^{-1}\sum _ jQ_ j$. Then $\omega _ q=N^{-1}\sum _ j\omega _{Q_ j}\geq \omega /2$ everywhere. Each point belongs to at least one smaller ball $\{ |z_ j|<r_ j\} $, so positivity of the other summands and (14.24.15) give

Thus (14.24.14) holds with $\varepsilon =(\tau /N)^ n$, independent of $\sup g$ and of all derivatives of $g$. $\square$