Theorem 10.8.18C34A54E553C3

Proof. If $Y=X$, take $s=f$, any $N$ and $C=1$. Suppose that $Y$ has codimension $q\geq 1$; if $f=0$, take $s=0$. As in the proof of theorem 10.7.9, we cut off a smooth extension of $f$ and correct the error, now with lemma 10.8.1; a pole of order $q$ along $Y$ in the weight makes the correction vanish on $Y$. Fix $R=R_ Y$ and the constant $b$ of lemma 10.8.8, with $\max _ XR<\mathrm e^{-4}$, and choose $b_ K\geq 0$ with $c_1(K_ X^{-1},h_{K^{-1},\omega })\geq -b_ K\omega $ (lemma 6.14.1). In Steps 1–4 we prove the theorem for a weight $\Phi $ with gentle analytic singularities, as for the approximations of theorem 6.17.6, under the curvature bound

In these steps $h$ denotes the metric $h_0\mathrm e^{-\Phi }$, and $A$ the pole set of $\Phi $. Step 5 treats arbitrary weights.

Step 1: the twisted curvature form. For the given $f$ choose $G$ and $\kappa $ by lemma 10.8.11. On $M=X\setminus (A\cup Y)$ consider $F=L\otimes K_ X^{-1}$ with the metric $h\otimes h_{K^{-1},\omega }\mathrm e^{-q\psi }$, where $\psi =\log R$; the canonical identification of $L$-valued functions with $F$-valued $(n,0)$-forms preserves the squared norms times the volume. The manifold $M$ carries a Kähler form and cutoffs as required in lemma 10.8.1, by lemma 10.7.5 applied to $\Phi +\psi $, which is quasi-plurisubharmonic, has pole set $A\cup Y$ and is smooth on $M$. The resulting Kähler form depends on $\Phi $, but the estimate of lemma 10.8.1 does not involve it.

Let $\psi _\varepsilon =\log (\varepsilon +R)$, $\mu $, $a_\varepsilon $ and $b_\varepsilon $ be the functions of lemma 10.8.4, applied on $M$ to the smooth function $R$, which satisfies $0<R<\mathrm e^{-4}$ there, for $0<\varepsilon <\mathrm e^{-4}$. For fixed $\varepsilon $, $a_\varepsilon $ and $b_\varepsilon $ are bounded above and below by positive constants, $a_\varepsilon \geq 1$ and $1\leq -\mu '(\psi _\varepsilon )\leq 2$. By part (2) of that lemma, applied with $2\pi $ times the curvature form of the metric of $F$,

Since $\mathrm i\partial \bar\partial \psi =2\pi \mathrm{dd}^{\mathrm{c}}\psi \geq -2\pi b\omega $, $0\leq R/(\varepsilon +R)\leq 1$ and $-\mu '\leq 2$, the identity (10.8.5) gives

where in the last step we used $a_\varepsilon \geq 1$ and $N_0-b_ K-qb=2b+2$. All constants in this display depend only on the geometry.

Step 2: the correction. Let $\rho \in C^\infty (\mathbb R)$ satisfy $0\leq \rho \leq 1$, $\rho =1$ on $(-\infty ,1/2]$ and $\rho =0$ on $[1,\infty )$, and put $G_\varepsilon =\rho (R/\varepsilon )G$. Since $\bar\partial R/\varepsilon =(1+R/\varepsilon )\bar\partial \psi _\varepsilon $, the $\bar\partial $-closed form $g_\varepsilon =\bar\partial G_\varepsilon $ splits as

regarded as $F$-valued $(n,1)$-forms. As in the proof of lemma 10.8.6, the rank-one part of (10.8.20) and lemma 10.7.1(1),(2) give

pointwise on the shell $\{ \varepsilon /2\leq R\leq \varepsilon \} $, where $R/\varepsilon \leq 1$, the norms being those of the metric of $F$. Hence the integral $M_{g_\varepsilon ^{(1)}}$ of lemma 10.8.1, formed with $H_\varepsilon $, is at most $C_1\int _{\{ \varepsilon /2\leq R\leq \varepsilon \} }|G|_ h^2R^{-q}\, \mathrm dV$, whose upper limit as $\varepsilon \to 0$ is at most $C_2\int _ Y|f|_ h^2\, \mathrm dV_ Y$ by lemma 10.8.15. Since $H_\varepsilon \geq 2\pi \omega $, lemma 10.7.1(3) bounds $M_{g_\varepsilon ^{(2)}}$ by

which tends to zero by (10.8.14).

Both forms are square integrable for the metric of $F$. For $g_\varepsilon ^{(1)}$, the derivative of the cutoff is bounded for fixed $\varepsilon $, and (10.8.13) bounds the integral of $|G|_ h^2R^{-q}$ over the shell, where the logarithm is bounded and $\mathrm e^{-\Phi }\leq \mathrm e^{\kappa \sup _ X\Phi }\mathrm e^{-(1+\kappa )\Phi }$; for $g_\varepsilon ^{(2)}$ this follows from (10.8.14). Since $M_{g'+g''}\leq 2(M_{g'}+M_{g''})$, by the Cauchy–Schwarz inequality for the positive form $\langle [H_\varepsilon ,\Lambda _\omega ]^{-1}\cdot ,\cdot \rangle $, lemma 10.8.1 gives $u_\varepsilon $ on $M$ with

The constants $C_1,\ldots ,C_4$ depend only on the geometry and the fixed cutoff $\rho $; neither the exponent $\kappa $ nor the derivatives of the local extensions $F_ i$ enter them.

Step 3: holomorphic sections with the trace $f$. For fixed $\varepsilon $ the denominator $a_\varepsilon +b_\varepsilon $ is bounded, and $h\geq \mathrm e^{-\sup _ X\Phi }h_0$ because $\Phi $ is bounded above on the compact manifold $X$. Hence $s_\varepsilon =G_\varepsilon -u_\varepsilon $, which is holomorphic on $M$, is square integrable for $h_0$, and it extends to a global holomorphic section by lemma 10.7.8. Then $u_\varepsilon =G_\varepsilon -s_\varepsilon $ is smooth on $X$.

It remains to show that $s_\varepsilon |_ Y=f$. Let $y\in Y\setminus A$, and choose holomorphic coordinates $(z',w)$ near $y$ with $Y=\{ w=0\} $. Near $y$ the weight $\Phi $ is smooth and $a_\varepsilon +b_\varepsilon $ is bounded, so $\int |u_\varepsilon |_{h_0}^2R^{-q}\, \mathrm dV$ is finite over a neighborhood of $y$. Since $R\leq C|w|^2$ near $y$ by lemma 10.8.8, and $\int _{|w|<r}|w|^{-2q}\, \mathrm dV(w)=\infty $ in $\mathbb C^ q$ for every $r>0$, by polar coordinates, the continuous function $u_\varepsilon $ vanishes at $y$. Hence $u_\varepsilon =0$ on $Y\setminus A$, and $s_\varepsilon =G=f$ there. Since the restriction of $h$ to $Y$ is defined, $A\cap Y$ is a proper analytic subset of the connected manifold $Y$, and the identity theorem gives $s_\varepsilon |_ Y=f$.

Step 4: the norm bound. For all sufficiently small $\varepsilon $,

because $a_\varepsilon +b_\varepsilon \leq 5\psi ^2$ by lemma 10.8.4(1), and because $R^ q(-\log R)^2$ is bounded on $X$. Consequently (10.8.21) bounds $\int _ X|u_\varepsilon |_ h^2\, \mathrm dV$ by $C_6$ times the integral there. The section $G$ has finite $h$-norm, since (10.8.13) controls it, $\Phi $ being bounded above and $R^ q(-\log R)^2$ bounded; therefore $\| G_\varepsilon \| _ h\to 0$ as $\varepsilon \to 0$, by absolute continuity of the integral on the shrinking supports. The triangle inequality gives

with $C_7=C_6C_4$. The sections $s_\varepsilon $ are bounded in $L^2$ for $h_0$, so, by lemma 10.6.2(2), a sequence $\varepsilon _ k\to 0$ gives sections converging uniformly to a holomorphic section $s$. Its restriction to $Y$ is $f$, and Fatou's lemma gives $\| s\| _ h^2\leq C_7\int _ Y|f|_ h^2\, \mathrm dV_ Y$. This proves the theorem for gentle weights with the constant $C_7$, which depends only on the geometry.

Step 5: arbitrary weights. Put $N=N_0+1$. By theorem 6.17.6, applied to $\varphi $ with the background $c_1(L,h_0)-N\omega $, there are $\Phi _ j\downarrow \varphi $ with gentle analytic singularities of the kind required in the gentle case and $c_1(L,h_0\mathrm e^{-\Phi _ j})\geq (N-\delta _ j)\omega $, where $\delta _ j\to 0$; for large $j$ the curvature is at least $N_0\omega $. Since $\Phi _ j\geq \varphi $, the restrictions of the metrics $h_0\mathrm e^{-\Phi _ j}$ to $Y$ are defined, and the norms of $f$ for them are at most its norm for $h$. The gentle case gives $s_ j\in H^0(X,L)$ with $s_ j|_ Y=f$ and $\int _ X|s_ j|_{h_0}^2\mathrm e^{-\Phi _ j}\, \mathrm dV\leq C_7\int _ Y|f|_ h^2\, \mathrm dV_ Y$. Since $\Phi _ j\leq \Phi _1$, the $s_ j$ are bounded in $L^2$ for $h_0$, and by lemma 10.6.2(2) a subsequence converges uniformly to a holomorphic section $s$ with $s|_ Y=f$. The weights $\mathrm e^{-\Phi _ j}$ converge to $\mathrm e^{-\varphi }$ wherever $\varphi $ is finite, hence almost everywhere, and Fatou's lemma gives (10.8.19) with $C=C_7$. $\square$