Theorem 6.5.4 (Ohsawa–Takegoshi)D315377B239D

Proof. After a normalization, Step 1 treats smooth weights on relatively compact pseudoconvex subsets, by correcting a cutoff of $f$ with lemma 6.5.2, lemma 6.5.3, and Step 2 regularizes $p$ and passes to the limit.

Normalization. The substitution $z_ n=4R\zeta $ maps $\mathbb C^{n-1}\times \Delta _{1/4}$ onto $\mathbb C^{n-1}\times \Delta _ R$, preserves the hyperplane and the integral $I$, and multiplies Lebesgue measure on $\mathbb C^ n$ by $16R^2$. Hence it suffices to show the following: if $\Omega \subset \mathbb C^{n-1}\times \Delta _{1/4}$, then $f$ has an extension $F\in \mathcal O(\Omega )$ with $\int _\Omega |F|^2\mathrm e^{-p}\, \mathrm dV\le C_0I$ for an absolute constant $C_0$; then $C=16C_0$ works. From now on we write the points of $\Omega $ as $(z',\zeta )$ with $z'\in \mathbb C^{n-1}$ and $|\zeta |<1/4$; for $n=1$ the variable $z'$ is absent. We may assume $\Omega _0\ne \varnothing $.

Step 1: smooth weights on a relatively compact set. Let $D\Subset \Omega $ be a pseudoconvex open set, let $U\subset \Omega $ be an open neighborhood of $\overline D$, and let $\varphi $ be a smooth plurisubharmonic function on $U$. We show that there are an open set $P\Subset \Omega _0$ with $\overline P\times \{ 0\} \subset U$ and a function $F\in \mathcal O(D)$ with $F(z',0)=f(z')$ whenever $(z',0)\in D$, such that

where $C_0$ is an absolute constant; we write $I_ P$ for the integral over $P$ on the right.

The set $K_0=\{ z':(z',0)\in \overline D\} $ is a compact subset of the open set $U_0=\{ z':(z',0)\in U\} \subset \Omega _0$; choose an open set $P$ with $K_0\subset P\Subset U_0$. There is $\varepsilon _0\in (0,1/4)$ such that every $(z',\zeta )\in D$ with $|\zeta |\le \varepsilon _0$ satisfies $z'\in P$: otherwise there would be points $(z'_ k,\zeta _ k)\in D$ with $\zeta _ k\to 0$ and $z'_ k\notin P$, and a subsequence would converge, $D$ being bounded, to a point $(z',0)\in \overline D$ with $z'\notin P$, which is impossible. Decreasing $\varepsilon _0$, we may also assume that $\overline P\times \overline\Delta _{\varepsilon _0}\subset U$ and, by the uniform continuity of $\varphi $ on this compact set, that

Fix a smooth function $\chi $ on $\mathbb R$ with $0\le \chi \le 1$, $\chi =1$ on $(-\infty ,1/4]$ and $\chi =0$ on $[1,\infty )$, for instance $\chi (x)=\kappa _0((4x-1)/3)$ with the function $\kappa _0$ fixed at the beginning of section 6.4, and put $c_\chi =\sup |\chi '|^2$. For $0<\varepsilon \le \varepsilon _0$ define $g_\varepsilon $ on $D$ by $g_\varepsilon (z',\zeta )=f(z')\chi (|\zeta |^2/\varepsilon ^2)$ if $z'\in P$, and $g_\varepsilon =0$ otherwise. The open sets $D\cap (P\times \mathbb C)$ and $D\cap \{ |\zeta |>\varepsilon \} $ cover $D$ by the choice of $\varepsilon _0$, and $g_\varepsilon $ is smooth on each of them (on the second it vanishes); hence $g_\varepsilon $ is smooth on $D$, and $g_\varepsilon (z',0)=f(z')$ for $(z',0)\in D$. Since $f$ is holomorphic,

is a $\bar\partial $-closed smooth form supported in $\{ \varepsilon /2\le |\zeta |\le \varepsilon \} $.

We solve $\bar\partial u=\alpha _\varepsilon $ on $D$ by lemma 6.5.2, with the function $\widetilde\varphi =\varphi +\log |\zeta |^2$ in place of $p$, which is plurisubharmonic on $D$ by proposition 5.10.3, proposition 5.9.5, and with $\psi =\psi _\varepsilon (\zeta )$ from lemma 6.5.3, a smooth bounded function on $D$ which is plurisubharmonic by proposition 5.9.3, applied to the projection $(z',\zeta )\mapsto \zeta $. The complex Hessian matrix $A=H_\psi $ has the single nonzero entry $(\psi _\varepsilon )_{\zeta \bar\zeta }$, so $|\bar\partial \psi |_{A^{-1}}^2$ is the function $h$ of lemma 6.5.3; thus $h<1$, and $h\le a_\varepsilon =2/(-\log (2\varepsilon ^2))$ on $\{ \alpha _\varepsilon \ne 0\} \subset \{ |\zeta |\le \varepsilon \} $. The dual energy of $\alpha _\varepsilon $ is $|f|^2|\chi '|^2|\zeta |^2\varepsilon ^{-4}/(\psi _\varepsilon )_{\zeta \bar\zeta }$, so the third estimate of lemma 6.5.3 and $|\zeta |\le \varepsilon $ give, on its support,

By (6.5.6) and Fubini's theorem, since every point of the support has $z'\in P$,

If $\varepsilon $ is so small that $a_\varepsilon \le 1/4$, then $(1+\sqrt{a_\varepsilon })/(1-\sqrt{a_\varepsilon })\le 3$, and lemma 6.5.2 gives $u_\varepsilon $ with $\bar\partial u_\varepsilon =\alpha _\varepsilon $ and $\int _ D(1-h)|u_\varepsilon |^2\mathrm e^{\psi _\varepsilon -\widetilde\varphi }\, \mathrm dV\le 3E$. By the first estimate of lemma 6.5.3, $(1-h)\mathrm e^{\psi _\varepsilon }\ge 1/(6\log ^2(|\zeta |^2+\varepsilon ^2))$, and $\mathrm e^{-\widetilde\varphi }=\mathrm e^{-\varphi }|\zeta |^{-2}$. Hence, with $C_1=225\pi c_\chi $,

On $D\cap \{ |\zeta |<\varepsilon /2\} $ we have $\bar\partial u_\varepsilon =0$, so $u_\varepsilon $ is holomorphic there, by lemma 6.4.10(1). Suppose that $u_\varepsilon (z'_0,0)\ne 0$ for some $(z'_0,0)\in D$. On a small polydisc around this point, $|u_\varepsilon |$ is bounded below by a positive constant, $\varphi $ is bounded above, and $\log ^2(|\zeta |^2+\varepsilon ^2)\le \log ^2(\varepsilon ^2)$; so the integrand in (6.5.7) is at least a positive constant times $|\zeta |^{-2}$, whose integral over a disc around $\zeta =0$ diverges. This contradicts (6.5.7). Hence $u_\varepsilon =0$ on $D\cap \{ \zeta =0\} $, and $F_\varepsilon =g_\varepsilon -u_\varepsilon $ is holomorphic on $D$, by the same lemma, because $\bar\partial F_\varepsilon =0$; moreover $F_\varepsilon (z',0)=f(z')$.

It remains to estimate $F_\varepsilon $. For $0<x=|\zeta |^2<1/16$ we have $x<x+\varepsilon ^2<1/8$, so $x\log ^2(x+\varepsilon ^2)\le x\log ^2x\le 4\mathrm e^{-2}<1$. Thus (6.5.7) gives $\int _ D|u_\varepsilon |^2\mathrm e^{-\varphi }\, \mathrm dV\le C_1I_ P$. By (6.5.6) and Fubini's theorem, $\int _ D|g_\varepsilon |^2\mathrm e^{-\varphi }\, \mathrm dV\le 2\pi \varepsilon ^2I_ P\le I_ P$. Therefore $\int _ D|F_\varepsilon |^2\mathrm e^{-\varphi }\, \mathrm dV\le 2(1+C_1)I_ P$, which is (6.5.5) with $C_0=2(1+C_1)$.

Step 2: the general case. By corollary 6.3.5 there are pseudoconvex open sets $D_1\subset D_2\subset \cdots $ with $D_ j\Subset \Omega $ and $\bigcup _ jD_ j=\Omega $. Choose $\varepsilon _ j\downarrow 0$ with $\varepsilon _ j<\operatorname {dist}(\overline D_ j,\mathbb C^ n\setminus \Omega )$, and let $\rho _\varepsilon $ be the radial mollifier of section 2.5; then, by proposition 5.9.2(2), $p_ j=p*\rho _{\varepsilon _ j}$ is a smooth plurisubharmonic function on the open neighborhood $U_ j=\{ z\in \Omega :\operatorname {dist}(z,\mathbb C^ n\setminus \Omega )>\varepsilon _ j\} $ of $\overline D_ j$, and $p_ j\downarrow p$ pointwise. Step 1, applied to $D_ j$, $U_ j$ and $\varphi =p_ j$, gives $F_ j\in \mathcal O(D_ j)$ with $F_ j(z',0)=f(z')$ for $(z',0)\in D_ j$ and

because $p_ j\ge p$. (On a component of $\Omega _0$ on which $p(\cdot ,0)\equiv -\infty $ we have $f=0$, and the last inequality holds with the convention $0\cdot \infty =0$.)

We show that a subsequence of $(F_ j)$ converges locally uniformly on $\Omega $. Let $K\subset \Omega $ be compact. Then $K\subset D_ k$ for some $k$, and for $j\ge k$ we have $p_ j\le p_ k$ on $K$, so $\int _ K|F_ j|^2\, \mathrm dV\le \mathrm e^{\max _ Kp_ k}C_0I$. It follows that $(F_ j)$ is uniformly bounded on compact subsets of $\Omega $. Indeed, for a holomorphic function $G$ the complex Hessian matrix of $|G|^2$ is $\bigl (\partial _{z_\mu }G\, \overline{\partial _{z_\nu }G}\bigr )_{\mu ,\nu }$, which is positive semidefinite; so $|G|^2$ is plurisubharmonic by proposition 5.9.2(1), and hence subharmonic by theorem 5.9.1. If $K\subset \Omega $ is compact, $\delta >0$ and $K_\delta =\{ z:\operatorname {dist}(z,K)\le \delta \} \subset \Omega $, then the mean value inequality over balls in theorem 3.4.5, applied to $|F_ j|^2$, gives $|F_ j(z)|^2\le |B(z,\delta )|^{-1}\int _{K_\delta }|F_ j|^2\, \mathrm dV$ for $z\in K$ and large $j$, and the right side is bounded independently of $j$ by the preceding estimate applied to $K_\delta $.

Next, let $K\subset \Omega $ be compact, and choose $r>0$ such that the compact set $K'$ of points at distance at most $2\sqrt n\, r$ from $K$ lies in $\Omega $. For $z\in K'_0=\{ z:\operatorname {dist}(z,K)\le \sqrt n\, r\} $, the closed polydisc of radius $r$ centered at $z$ lies in $K'$, and the coefficient bound in (5.2.1) with $a=z$ gives $|\partial _{z_ l}F_ j(z)|\le r^{-1}\max _{K'}|F_ j|$ for $1\le l\le n$ and all large $j$ (those with $K'\subset D_ j$). Since $\mathrm dF_ j=\sum _ l\partial _{z_ l}F_ j\, \mathrm dz_ l$, it follows that these $F_ j$ are Lipschitz with a common constant on every ball of radius $\sqrt n\, r$ centered at a point of $K$, so the sequence $(F_ j)$ is equicontinuous on $K$ from some index on. By proposition 2.7.4 on a compact exhaustion of $\Omega $ and a diagonal argument, a subsequence converges locally uniformly on $\Omega $ to a continuous function $F$. Since $\bar\partial F_ j=0$ and $F_ j\to F$ in the sense of distributions, $\bar\partial F=0$, and $F$ is holomorphic by lemma 6.4.10(1). Moreover $F(z',0)=\lim _ jF_ j(z',0)=f(z')$ for $z'\in \Omega _0$.

Finally, let $K\subset \Omega $ be compact, and let $k$ be so large that $K\subset D_ k$. For $j\ge k$ we have $p_ j\le p_ k$ on $K$, so locally uniform convergence gives

This holds for all large $k$, and $\mathrm e^{-p_ k}$ increases to $\mathrm e^{-p}$ as $k\to \infty $; so the monotone convergence theorem gives $\int _ K|F|^2\mathrm e^{-p}\, \mathrm dV\le C_0I$, and exhausting $\Omega $ by compact sets gives $\int _\Omega |F|^2\mathrm e^{-p}\, \mathrm dV\le C_0I$. $\square$