Theorem 14.14.1.2763FB514B34 Fix smooth Hermitian forms $\gamma _0,\gamma $ on a compact complex manifold $X$ of dimension $n\geq 2$. For a smooth real function $h$ on $X$, suppose that $v$ is a smooth solution of
\[ Q=Q_0+P_\gamma (\mathrm i\partial \bar\partial v)+Z(\partial v)>0, \qquad \log \det _\gamma Q=h, \]
with the tensors of section 14.13. If $\| v\| _{C^0}$ and $\| h\| _{C^2}$ are bounded, then
\[ \sup _ X|\mathrm i\partial \bar\partial v|_\gamma \leq C\bigl (1+\sup _ X|\partial v|_\gamma ^2\bigr ), \]
where $C$ depends only on these bounds and the fixed smooth data.
Proof.
Adding a constant, we may assume $\sup _ Xv=0$, and we write $-M\leq v\leq 0$. Set
\[ K=1+\sup _ X|\partial v|_\gamma ^2,\qquad \phi (s)=-\tfrac 12\log (1-s/(2K)),\qquad \psi (t)=D_1\mathrm{e}^{-D_2t}. \]
The positive constants $D_1,D_2$ will be chosen below using only the fixed data. On $0\leq s\leq K$,
\[ \frac1{4K}\leq \phi '\leq \frac1{2K},\qquad \phi ''=2(\phi ')^2,\qquad 0\leq \phi \leq \tfrac 12\log 2. \]
Let $\lambda _1$ be the largest eigenvalue of $g=\chi +\mathrm i\partial \bar\partial v+W$ relative to $\gamma $. It is positive, because $\operatorname {tr}_\gamma g=\operatorname {tr}_\gamma Q>0$. Consider a maximum point of
\[ H=\log \lambda _1+\phi (|\partial v|_\gamma ^2)+\psi (v). \]
It suffices to bound $\lambda _1/K$ at this point, because the other two terms of $H$ are uniformly bounded once $D_1,D_2$ are fixed. We may assume that this ratio exceeds any fixed constant specified in the course of the argument.
Choose holomorphic coordinates that are unitary at the maximum point, with $g$ diagonal there and eigenvalues ordered as $\lambda _1\geq \cdots \geq \lambda _ n$. All estimates below are at this point, and repeated indices are summed only when a sum is written. Put $\lambda =\lambda _1$ and use the weights $t_ i,f_ k$ and their sum $\mathcal F$ from lemma 14.13.5. Since $\sum _ i\lambda _ i>0$ and the eigenvalues are ordered,
\[ |\lambda _ i|\leq (n-1)\lambda ,\qquad |v_{i\bar j}|\leq C\lambda \quad \hbox{if }\lambda \geq K. \]
The constant $C$ may change from line to line; it is independent of $K$ and, until $D_1,D_2$ are fixed below, also of $D_1,D_2$. The pure second derivatives $v_{ij}$ are not covered by this bound.
$\square$
Pome
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