Lemma 12.27.21C23EA256441

Proof. Step 1: a first identification. Start with the normal exponential map of a smooth Hermitian metric near $D$. Its differential along $D$ is complex linear and induces the identity on the normal quotient. In a parallel local frame and a local coordinate $z_0$ on $D$ with $\vartheta _ D=\mathrm dz_0$, choose holomorphic coordinates $Z,W$ on $S$ with $D=\{ W=0\} $. Their pullbacks under the first map satisfy

The residue normalization gives $\Omega _ S=(1+Wa(Z,W))\, \mathrm dW/W\wedge \mathrm dZ$ with $a$ holomorphic. Substituting, and noting that $\Theta \wedge \mathrm dw=w\Theta \wedge \Theta =0$, we obtain $\Omega _ S-\Omega _0=O(|w|)$ in the coframe $\mathrm dz_0,\mathrm d\bar z_0,\Theta ,\bar\Theta $. Every derivative by $\partial _{z_0},\partial _{\bar z_0},w\partial _ w,\bar w\partial _{\bar w}$ preserves this bound. Since $|w|=\mathrm e^{-z/2}$, this proves the cylindrical estimate for the first map.

Step 2: holomorphic fiber disks. It remains to modify the first map so that the fiber disks become holomorphic, without changing its first jet along $D$. On a fiber disk $\Delta _{p,\rho }\subset N_ p$ of radius $\rho $, perturb the map by $\Phi _ Y(q)=\exp _ q(Y_ q)$, where $Y$ is a vector field along the image of the disk. Identify the target tangent spaces with one fixed complex vector space over $p$, by Chern parallel transport along the radial paths and the short exponential paths. In these identifications, the antiholomorphic part of the differential of the perturbed disk is

Fix a Hölder exponent $0<\nu <1$, and use $C^{k,\nu }$ norms on the disks rescaled to unit size. The Cauchy–Green operator $T_\rho $ of the disk gives the normalized right inverse

of $\bar\partial $. We restrict its domain to $(0,1)$-forms vanishing at the center; its image consists of fields whose value and first jet vanish at the center, and on this image $\bar\partial $ and $R_\rho $ are inverse to each other. The differential of $\mathcal F_{p,\rho }$ at zero has the form $\bar\partial +K_{p,\rho }$, with

Indeed, the two complex structures agree along the zero section, and rescaling by $w\mapsto \rho w$ contributes an additional factor $\rho $ to a pulled-back one-form. For small $\rho $, the operator $R_\rho (I+K_{p,\rho }R_\rho )^{-1}$ is a right inverse of the differential, and the quantitative implicit function theorem gives a correction $Y=O(|w|^2)$ with vanishing first jet that makes the disk holomorphic.

Step 3: global smoothness. Compactness of $D$ makes the choice of $\rho $ uniform in $p$. The normalized Cauchy operator is equivariant under unitary changes of the fiber frame, so the intrinsic parallel transports and the uniqueness of the small solution make the corrections agree on overlaps. The parametric implicit function theorem gives smooth dependence on $p$, and interior elliptic regularity on slightly smaller fiber disks gives smoothness along the fibers. Since the first jet is unchanged, the correction changes the map by $O(|w|^2)$ together with its cylindrical derivatives. Composing the first map with this correction therefore preserves the estimate and proves the lemma. $\square$