Lemma 10.8.6833820F8A53D

Proof. If $f=0$, take $F=0$. Put $n=m+q$, and let $\beta $ be the Euclidean Kähler form on $\mathbb C^ n$. We first treat smooth weights: we cut off the trivial extension of $f$ near $\{ w=0\} $, correct it with lemma 10.8.1 on $D\setminus \{ w=0\} $, and let the width $\varepsilon $ of the cutoff tend to zero. So assume that $p$ is smooth near $\overline D$, and set $R=|w|^2$, $\psi =\log R$ and $\psi _\varepsilon =\log (\varepsilon +R)$ for small $\varepsilon >0$. The function $P=p+\pi |z|^2$, where $z=(z',w)$, satisfies $\mathrm{dd}^{\mathrm{c}}P\geq \beta $, and replacing $p$ by $P$ changes both sides of (10.8.7) by factors bounded in terms of $r'$ and $r''$ only; we prove the estimate for $P$.

The twisted form. On $D\setminus \{ w=0\} $ consider the trivial line bundle with the metric $\mathrm e^{-P-q\psi }$, and let $\mu $, $a_\varepsilon $ and $b_\varepsilon $ be the functions of lemma 10.8.4, for $0<\varepsilon <\mathrm e^{-4}$; that lemma applies because $0<R<\mathrm e^{-4}$ on $D\setminus \{ w=0\} $, since $r''<\mathrm e^{-2}$. By its part (1), $a_\varepsilon \geq 1$ and $-\mu '(\psi _\varepsilon )\geq 1$, and for fixed $\varepsilon $ the functions $a_\varepsilon $ and $b_\varepsilon $ are bounded above and below by positive constants. By its part (2), applied with $\Theta =\mathrm i\partial \bar\partial (P+q\psi )$, the curvature form of this metric times $2\pi $, the gradient terms cancel in the form $H$ of lemma 10.8.1, and

Both logarithmic Hessians $\mathrm i\partial \bar\partial \psi $ and $\mathrm i\partial \bar\partial \psi _\varepsilon $ are semipositive, so (10.8.5) gives $\mathrm i\partial \bar\partial \psi _\varepsilon \geq (\varepsilon /R)\, \mathrm i\partial \psi _\varepsilon \wedge \bar\partial \psi _\varepsilon $. Since also $\mathrm i\partial \bar\partial P=2\pi \mathrm{dd}^{\mathrm{c}}P\geq 2\pi \beta $, the bounds $a_\varepsilon \geq 1$ and $-\mu '(\psi _\varepsilon )\geq 1$ give $H_\varepsilon \geq 2\pi \beta +(\varepsilon /R)\, \mathrm i\partial \psi _\varepsilon \wedge \bar\partial \psi _\varepsilon $.

Complete cutoffs. The manifold $D\setminus \{ w=0\} $ carries a Kähler form and cutoffs as required in lemma 10.8.1. Indeed, put $\tau _0=-\log (r'^2-|z'|^2)-\log (r''^2-|w|^2)$, omitting the first term if $m=0$, and $\tau _1=-\log (-\psi )$. Each of the three functions $\tau $ formed by the two terms of $\tau _0$ and by $\tau _1$ is of the form $\tau =-\log g$ with $g>0$ smooth and $-\mathrm{dd}^{\mathrm{c}}g\geq 0$, namely $g=r'^2-|z'|^2$, $g=r''^2-|w|^2$ and $g=-\psi $; hence

Thus $\tau _0$ and $\tau _1$ are plurisubharmonic, and $\sigma =\beta +\mathrm{dd}^{\mathrm{c}}\tau _0+\mathrm{dd}^{\mathrm{c}}\tau _1\geq \beta $ is a Kähler form. Moreover $\sigma \geq \mathrm{dd}^{\mathrm{c}}\tau \geq \frac{\mathrm i}{2\pi }\partial \tau \wedge \bar\partial \tau $ for each of the three functions; the eigenvalue of the rank-one form $\frac{\mathrm i}{2\pi }\partial \tau \wedge \bar\partial \tau $ with respect to $\sigma $ is $|\partial \tau |_\sigma ^2/(2\pi )$, so $|\partial \tau |_\sigma ^2\leq 2\pi $, and these functions have bounded gradients for $\sigma $. The function $\tau _0-\tau _1$ tends to $+\infty $ at the boundary of $D$ and along $\{ w=0\} $, and its sublevel sets are compact in $D\setminus \{ w=0\} $. If $\lambda \in C^\infty (\mathbb R)$ equals one on $(-\infty ,1]$ and zero on $[2,\infty )$, the functions $\chi _ j=\lambda ((\tau _0-\tau _1)/j)$ are compactly supported, tend to one, and have gradients $O(1/j)$. The canonical bundle of $\mathbb C^ n$ is trivialized by $\mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n$, whose norm for $\beta $ is constant, so we identify functions with $(n,0)$-forms, as in the proof of theorem 10.7.9; this multiplies all the pointwise norms below by the same constant factor.

The correction. Let $G(z',w)=f(z')$. Choose $\rho \in C^\infty (\mathbb R)$ with $0\leq \rho \leq 1$, equal to one on $(-\infty ,1/2]$ and to zero on $[1,\infty )$, and put $G_\varepsilon =\rho (R/\varepsilon )G$. Since $\bar\partial R/\varepsilon =(1+R/\varepsilon )\bar\partial \psi _\varepsilon $,

which is supported in $\{ \varepsilon /2\leq R\leq \varepsilon \} $. For an $(n,1)$-form $v$, the rank-one part of the lower bound for $H_\varepsilon $ and lemma 10.7.1(1),(2), applied with $\alpha =\partial \psi _\varepsilon $, give

where $|G|$ is the modulus of the function $G$ and the inner products are those of $\beta $ and the metric $\mathrm e^{-P-q\psi }=\mathrm e^{-P}R^{-q}$. Hence, on the support of $g_\varepsilon $, where $R/\varepsilon \leq 1$, we have $\langle [H_\varepsilon ,\Lambda _\beta ]^{-1}g_\varepsilon ,g_\varepsilon \rangle \leq C|G|^2\mathrm e^{-P}R^{-q}$, and the integral $M_{g_\varepsilon }$ of lemma 10.8.1 is at most

In polar coordinates in $w$, the integral of $R^{-q}$ over the shell $\varepsilon /2\leq |w|^2\leq \varepsilon $ is a fixed multiple of $\int _{\sqrt{\varepsilon /2}}^{\sqrt\varepsilon }\mathrm dt/t=\frac12\log 2$. Since $P$ is uniformly continuous on $\overline D$ and $G$ does not depend on $w$, the last integral tends to a fixed multiple of $\int _{B'}|f|^2\mathrm e^{-P(z',0)}\, \mathrm dV_{B'}$ as $\varepsilon \to 0$. Lemma 10.8.1 gives $u_\varepsilon $ on $D\setminus \{ w=0\} $ with $\bar\partial u_\varepsilon =g_\varepsilon $ and

For fixed $\varepsilon $ the denominator is bounded and $\mathrm e^{-P}R^{-q}$ is bounded below, so $u_\varepsilon $ is square integrable, and $F_\varepsilon =G_\varepsilon -u_\varepsilon $, which is holomorphic off $\{ w=0\} $, extends holomorphically to $D$ by lemma 10.7.8. The difference $u_\varepsilon =G_\varepsilon -F_\varepsilon $ is then smooth near $\{ w=0\} $, and it vanishes there: a nonzero value at a point $(z'_0,0)$ would make $\int |u_\varepsilon |^2|w|^{-2q}$ diverge near that point, since $\int |w|^{-2q}\, \mathrm dV(w)$ diverges logarithmically at $w=0$ in $\mathbb C^ q$. Hence $F_\varepsilon (z',0)=G_\varepsilon (z',0)=f(z')$.

The limit $\varepsilon \to 0$. By lemma 10.8.4(1), $a_\varepsilon +b_\varepsilon \leq 5(-\log (\varepsilon +R))^2$, so the correction estimate bounds $u_\varepsilon $ in the weaker weight $(\varepsilon +R)^{-q}(-\log (\varepsilon +R))^{-2}\mathrm e^{-P}$. In this weight the squared norm of $G_\varepsilon $ is at most $C_ P(-\log \varepsilon )^{-2}$, because its support has normal volume $O(\varepsilon ^ q)$ and $(\varepsilon +R)^{-q}\leq \varepsilon ^{-q}$ there; the constant $C_ P$ is finite for the fixed smooth weight $P$. Consequently the squared norms of $F_\varepsilon $ in this weight have upper limit, as $\varepsilon \to 0$, at most a constant, depending only on $m,q,r',r''$, times $\int _{B'}|f|^2\mathrm e^{-P(z',0)}$. Since $t^{-q}(-\log t)^{-2}$ is bounded below on $(0,\mathrm e^{-3}]$, the $F_\varepsilon $ are bounded in $L^2(D)$, and by lemma 10.6.2(1) a sequence $\varepsilon _ k\to 0$ gives functions converging locally uniformly to a holomorphic $F$ with $F(z',0)=f(z')$. By Fatou's lemma, $F$ satisfies (10.8.7) with $P$ in place of $p$, and removing the bounded factor $\mathrm e^{-\pi |z|^2}$ gives (10.8.7) for smooth $p$.

General $p$. Take smooth plurisubharmonic functions $p_ l\downarrow p$ on a neighborhood of $\overline D$, obtained by convolution: by (3.4.13) the convolutions $p*\rho _{1/l}$ decrease to $p$, they are smooth, and they are plurisubharmonic because $\mathrm{dd}^{\mathrm{c}}(p*\rho _{1/l})=(\mathrm{dd}^{\mathrm{c}}p)*\rho _{1/l}\geq 0$ (theorem 5.9.1). Their restrictions $p_ l(z',0)$ decrease to $p(z',0)$, so the right sides of (10.8.7) for $p_ l$ are at most the right side for $p$, and the smooth case gives extensions $F_ l$ satisfying (10.8.7) for $p_ l$ with the same constant. Since $p_ l\leq p_1$, the weights have a common positive lower bound, so the $F_ l$ are bounded in $L^2(D)$ and, by lemma 10.6.2(1), a subsequence converges locally uniformly on $D$, including on $\{ w=0\} $. The limit $F$ satisfies $F(z',0)=f(z')$, and Fatou's lemma gives (10.8.7). $\square$