Lemma 12.9.6.9F57BA26805A Let $F$ be smooth with $\| F-F_*\| _{C^{2,\alpha }}\leq \varepsilon $. The solution form $\Omega _ F=\omega +\mathrm{dd}^{\mathrm{c}}u_ F$ of $\mathrm{MA}_\omega (u_ F)=\mathrm{e}^{B_0(F)+F}\mathrm{d}V$ and its Gauduchon factor $\chi _ F$, normalized as below, satisfy
\begin{gather*} \tfrac 12\omega \leq \Omega _ F\leq \tfrac 32\omega , \qquad \frac m2\leq \chi _ F\leq \frac{3a}2,\\ \mathrm{dd}^{\mathrm{c}}(\chi _ F\Omega _ F^{n-1})=0, \qquad \int _ X\chi _ F\omega ^ n=1. \end{gather*}
The potential can be normalized by $\int _ Xu_ F\, \mathrm{d}V=0$ with $\| u_ F\| _{C^{4,\alpha }}+|B_0(F)-\log V|\leq r$.
Proof.
For $\| v\| _{C^{4,\alpha }}\leq 1/(2D)$ put
\[ N(v)=\log \det (I+\omega ^{-1}\mathrm{dd}^{\mathrm{c}}v)-L_\omega v. \]
The logarithm series starts at degree two. If the endomorphism norms of $\omega ^{-1}\mathrm{dd}^{\mathrm{c}}v$ and $\omega ^{-1}\mathrm{dd}^{\mathrm{c}}w$ are at most $\rho \leq 1/2$, telescoping their powers bounds the difference series by $C_{\mathrm{tr}}\rho /(1-\rho )$ times the endomorphism difference. Hence
\begin{align*} \| N(v)\| _{C^{2,\alpha }}& \leq Q\| v\| _{C^{4,\alpha }}^2,\\ \| N(v)-N(w)\| _{C^{2,\alpha }} & \leq Q(\| v\| _{C^{4,\alpha }}+\| w\| _{C^{4,\alpha }}) \| v-w\| _{C^{4,\alpha }}. \end{align*}
On the closed radius-$r$ ball of normalized pairs $(v,c)$, the map
\[ (v,c)\longmapsto \mathcal L^{-1}(F-F_*-N(v)) \]
has image norm at most $K\varepsilon +KQr^2\leq 3r/4$ and Lipschitz constant at most $2KQr\leq 1/2$. Its fixed point $(u,c)$ satisfies
\[ \log \frac{(\omega +\mathrm{dd}^{\mathrm{c}}u)^ n}{\omega ^ n}-c=F-F_*. \]
The estimate $Dr\leq 1/2$ gives the form comparison. The scalar is $b=\log V+c$. Differentiating the local equation gives a uniformly elliptic equation for each first derivative of $u$, with coefficients initially in $C^{2,\alpha }$. Interior Schauder estimates improve the regularity one order at a time, so $u$ is smooth. Uniqueness in theorem 14.2.1 identifies it with $u_ F$ and gives $b=B_0(F)$. No uniform higher-derivative bounds on $F$ enter the estimates above.
Put $\Delta A=A_{\Omega _ F}-A_\omega $. Stokes' theorem makes its image lie in $Z_0$. On the complete space $E_\chi $, the affine map
\[ \eta \longmapsto -\mathcal R\Delta A(\chi _0+\eta ) \]
has Lipschitz constant $\kappa =GC_ ADr\leq m/(4C_0)\leq 1/4$. Its fixed point obeys
\[ \| \eta \| _{C^{2,\alpha }} \leq \frac{\kappa C_0}{1-\kappa } \leq 2\kappa C_0\leq m/2. \]
Since $A_\omega \chi _0=0$ and $A_\omega \mathcal R$ is the identity on $Z_0$, the function $\chi _ F=\chi _0+\eta $ satisfies $A_{\Omega _ F}\chi _ F=0$, and it has the stated normalization and bounds. Elliptic regularity makes it smooth. Every Gauduchon factor of $\Omega _ F$ with the same normalization satisfies the same affine fixed-point equation, so this factor is the one given by theorem 11.4.2.
$\square$
Pome
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