Theorem 12.27.358A77103A732

Proof. We look for a fiber-preserving change of the identification of lemma 12.27.2 after which the coordinate $z$ becomes pluriharmonic. The logarithm of the flat norm of the defining section is then $-z$, up to a pluriharmonic correction.

Step 1: horizontal forms. Use the identification $\Upsilon $ of lemma 12.27.2, and omit the pullback notation. A form on $N^\times $ is horizontal if its contractions with $\xi $ and $\bar\xi $ vanish. For a horizontal form $\eta $ define

A form $\eta _ m$ on $D$ with values in $N^{-m}$ defines the horizontal form $w^ m\eta _ m$, homogeneous of degree $m$ in $w$, and direct differentiation shows that $\mathrm d_ H$ acts on it as the flat covariant exterior derivative. Since the fibers are holomorphic,

with $\alpha ,\beta $ horizontal. Closedness gives $\mathcal L_{\bar\xi }\alpha =\mathcal L_{\bar\xi }\beta =0$ and $\mathcal L_\xi \beta =\mathrm d_ H\alpha $.

Step 2: a horizontal inverse. Fix $0<\delta <1/2$, and let $U_ T=\{ z\geq T\} $. For tensors define the weighted norms

Let $\mathcal H^{k+1}_{\delta ,T}$ be the space of functions $f$ on $U_ T$ with $\| f\| _{k+1,\delta ,T}<\infty $ and $\mathcal L_{\bar\xi }f=0$, and let $\mathcal D^ k_{\delta ,T}$ be the space of complex two-forms $\gamma $ on $U_ T$ with $\| \gamma \| _{k,\delta ,T}<\infty $, $\mathrm d\gamma =0$ and $\iota _{\bar\xi }\gamma =0$. We construct an operator

with constant independent of $T$, such that

Write $\gamma =\Theta \wedge \alpha +\beta $, with $\alpha ,\beta $ horizontal. The conditions on $\gamma $ give $\mathcal L_{\bar\xi }\gamma =0$ and, in the expansion in homogeneous components along the fibers, $m\beta _ m=\mathrm d_ H\alpha _ m$. Only components with $m\geq 1$ occur, because the weighted norm is finite and $\delta >0$ excludes the Laurent components of degree $m\leq 0$. Decompose

Lemma 12.27.1 gives sections $e_ m$ with $\bar\partial e_ m=\iota _ v\partial v_ m$ and $\| e_ m\| _{H^ k}\leq \| v_ m\| _{H^ k}$. Put $f_ m=-(u_ m+e_ m)/m$. Then

Also $\| \bar\partial f_ m\| _{H^ k}\leq C\| \beta _ m\| _{H^ k}$. Since $D$ is flat, integration by parts gives the same bound for $\partial f_ m$, which controls the horizontal derivatives of order $k+1$. A vertical derivative of a homogeneous component multiplies it by $m$, so every derivative involving a vertical direction is controlled by the bound for $mf_ m$. The components are orthogonal on the circles $\{ |w|=\text{constant}\} $, and the squared radial factor of the $m$th component is $\mathrm e^{-mz}$ with $m-2\delta >0$; integrating the weight gives the required estimate for $f=\sum _{m\geq 1}f_ m$, where each $f_ m$ is identified with its homogeneous horizontal lift, and we put $Q_ T\gamma =f$.

For forms with Sobolev coefficients, regularize each component by the heat operator of the flat base, which commutes with the differential equations above. The uniform estimates and orthogonality make the finite partial sums converge in the target space, and passing to the limit proves the first identity in (12.27.4). If $\gamma =\mathrm df\wedge \vartheta $, then $\alpha _ m=mf_ m\vartheta $, so $v_ m$ and the normalized $e_ m$ vanish, and the construction returns $f_ m$. This proves the second identity.

Step 3: a fixed point. Put $\eta =\Omega _ S-\Omega _0$, and choose an integer $k\geq 4$. By the cylindrical estimate, the weighted $H^{k+2}$ norm of $\eta $ on $U_{T-1}$ and its first $k+2$ derivatives are bounded by a number $a_ T$ with $a_ T\to 0$. For small $f\in \mathcal H^{k+1}_{\delta ,T}$, the map $\Phi _ f(\zeta )=\mathrm e^{f(\zeta )}\zeta $, which rescales each fiber point $\zeta $, is a $C^2$ diffeomorphism onto its image and maps $U_ T$ into $U_{T-1}$. Its restriction to every fiber is holomorphic, so $\Phi _ f^*\Omega _ S-\Omega _0$ lies in the domain of $Q_ T$. We look for $f$ such that $\Phi _ f^*\Omega _ S$ has no horizontal component. Since

(12.27.4) reduces this to the equation

On a fixed sufficiently small ball, multiplication and composition estimates in weighted Sobolev spaces on the flat cylinder give

The constants are independent of $T$, because translation in the cylindrical direction is an isometry, the weight is normalized at $T$, and the shifts of $z$ are uniformly bounded. Differentiating the pullback along the segment from $f$ to $g$ involves one Lie derivative of the fixed form $\eta $ and at most one derivative of $f-g$, and the gain of one derivative in $Q_ T$ restores the norm. For $T$ sufficiently large, $-E_ T$ is therefore a contraction of the closed ball of radius $2Ca_ T$ into itself. Let $f$ be its fixed point.

Step 4: a pluriharmonic function. By the first identity in (12.27.4), the horizontal component of $\Phi _ f^*\Omega _ S$ vanishes, so $\Theta \wedge \Phi _ f^*\Omega _ S=0$. The pulled-back two-form is nowhere zero and of type $(2,0)$ for $\Phi _ f^*J_ S$, so $\Theta $ is of type $(1,0)$ for this complex structure. Since $\mathrm dz=-(\Theta +\bar\Theta )$ and $\mathrm d\Theta =0$, the function $z$ is pluriharmonic for $\Phi _ f^*J_ S$.

Step 5: the flat metric. Return to $S$ through $A=\Upsilon \circ \Phi _ f$. On each fiber, $f$ is bounded and holomorphic on a punctured disk, so it extends holomorphically across the center, and its expansion in positive powers of $w$ gives $f(0)=0$. Since $f$ is uniformly small, Rouché's theorem, comparing $w\mathrm e^{f(w)}-c$ with $w-c$ on a fixed smaller circle, shows that every sufficiently small value $c$ is attained. Hence the image of $A$ contains a punctured neighborhood of $D$. The function $r=z\circ A^{-1}$ is $C^2$ and pluriharmonic there, hence smooth by elliptic regularity. If $w_ S$ is a local holomorphic defining function for $D$, the unchanged first normal jet of $\Upsilon $ and the bounded factor $\mathrm e^ f$ give

Thus $r+\log |w_ S|^2$ is bounded and pluriharmonic off $D$. Both this function and its negative are bounded plurisubharmonic functions off $D$, so they extend plurisubharmonically across $D$; the sum of the extensions vanishes almost everywhere, hence everywhere, so the extension is pluriharmonic. It follows that $F=-r$, which has a logarithmic pole along $D$, satisfies $\mathrm{dd}^{\mathrm{c}}F=[D]$ near $D$. Locally, the metric for which the squared norm of the defining section equals $\mathrm e^{F}$ is smooth and flat, since $F-\log |w_ S|^2$ is smooth and pluriharmonic. These local metrics agree on overlaps and give the required unitary flat metric. $\square$