Theorem 10.7.9846B917F28CD

Proof. If $v=0$, take $s=0$; assume $v\neq 0$. We cut off a local extension of $v$ and correct the error with lemma 10.7.6 outside $x$ and the poles of gentle approximations of $\varphi $; a logarithmic pole at $x$ in the weight makes the correction vanish at $x$.

Step 1: data independent of $x$. Choose finitely many coordinate charts and, inside them, balls $B_ k$ of radii $\varrho _ k$, with closures in the charts, whose concentric balls of half the radius cover $X$. Put $r_0=\min _ k\varrho _ k/8$. Then for every $x\in X$ there is $k$ with $x\in \frac12B_ k$, and the coordinate ball $B$ of radius $4r_0$ centered at $x$ in the $k$th chart lies in $B_ k$. Let $\lambda $ be a smooth function on $\mathbb R^{2n}$ with values in $[0,1]$, equal to one on the ball of radius $2r_0$ and zero outside the ball of radius $3r_0$, and put

where $z$ are the chart coordinates. Then $\psi _ x\leq 0$, $\psi _ x$ is plurisubharmonic where $\lambda (z-z(x))=1$, and on the annulus where $\lambda (z-z(x))$ is not constant the logarithm and its first two derivatives are bounded independently of $x$. Hence there is $b_0\geq 0$, independent of $x$, with

Choose $b_1\geq 0$ with $c_1(K_ X^{-1},h_{K^{-1},\omega })\geq -b_1\omega $ (lemma 6.14.1), and put $N=b_0+b_1+2$.

Step 2: a local extension and its cutoff. By lemma 10.6.1, $L$ has a holomorphic frame $e$ over $B$. With $|e|_{h_0}^2=\mathrm e^{-p_0}$, the local weight $p_0+\varphi $ of $h$ is plurisubharmonic on $B$, because $\mathrm{dd}^{\mathrm{c}}(p_0+\varphi )=c_1(L,h)\geq 0$. By corollary 6.5.8, applied in this frame on the ball of radius $r_0$ around $x$, whose diameter $2r_0$ is fixed, there is a holomorphic section $f$ of $L$ over this ball with $f(x)=v$ and

here $C_0$ does not depend on $x$, because the volume densities are uniformly comparable on the finite cover. Let $\chi $ be a cutoff equal to one for $|z-z(x)|<r_0/2$ and zero for $|z-z(x)|>3r_0/4$, obtained by translating a fixed one. Put $G=\chi f$, extended by zero, and $g=\bar\partial G=f\, \bar\partial \chi $. The form $g$ is supported in the annulus $r_0/2\leq |z-z(x)|\leq 3r_0/4$, on which $\psi _ x$ and $\bar\partial \chi $ are bounded independently of $x$.

Step 3: global sections for the approximations of $\varphi $. Apply theorem 6.17.6 to $\varphi $ with the background $c_1(L,h_0)-N\omega $. It gives functions $\varphi _ j\downarrow \varphi $ with gentle analytic singularities, analytic pole sets $A_ j$, and $c_1(L,h_0)+\mathrm{dd}^{\mathrm{c}}\varphi _ j\geq (N-\varepsilon _ j)\omega $ with $\varepsilon _ j\to 0$. Since $\varphi _ j(x)\geq \varphi (x)>-\infty $, no $A_ j$ contains $x$. Put

For $j$ so large that $\varepsilon _ j\leq 1$, this metric is smooth on $M_ j=X\setminus (A_ j\cup \{ x\} )$ and its curvature is at least $(N-\varepsilon _ j-b_0-b_1)\omega \geq \omega $. By lemma 10.7.5, applied to $\varphi _ j+\psi _ x$, the manifold $M_ j$ carries a Kähler form $\sigma \geq \omega /2$ and cutoffs as required in lemma 10.7.6. The canonical identification of $L$-valued $(0,q)$-forms with $F$-valued $(n,q)$-forms preserves the pointwise norms for $\omega $, because the $(n,0)$-covector and its dual cancel. Under this identification $g$ is $\bar\partial $-closed, and since $\varphi _ j\geq \varphi $,

with $C_1$ depending only on the fixed cutoffs and the bounds for $\psi _ x$ on the annulus. Lemma 10.7.6, with $a=1$, gives $u_ j$ on $M_ j$ with $\bar\partial u_ j=g$ and

The section $s_ j=G-u_ j$ is holomorphic on $M_ j$. The weights $\mathrm e^{-\varphi _ j-\psi _ x}$ are bounded below by a positive constant, since $\varphi _ j\leq \varphi _1$ is bounded above and $\psi _ x\leq 0$; hence $s_ j$ is square integrable for $h_0$ near $A_ j\cup \{ x\} $, and lemma 10.7.8 extends it to a global holomorphic section. Near $x$ we have $g=0$ and no point of $A_ j$, so $u_ j=G-s_ j$ is holomorphic near $x$. Moreover $|u_ j|_{h_0}^2\mathrm e^{-\varphi _ j-\psi _ x}\geq c\, |u_ j|_{h_0}^2|z-z(x)|^{-2n}$ near $x$ for a constant $c>0$, by the upper bound for $\varphi _ j$ and since $\psi _ x=n\log |z-z(x)|^2+O(1)$. Since $\int _{|z-z(x)|<r}|z-z(x)|^{-2n}\, \mathrm dV=\infty $ for every $r>0$, by polar coordinates in $\mathbb R^{2n}$, the finiteness of the weighted integral of the continuous function $u_ j$ forces $u_ j(x)=0$. Hence $s_ j(x)=G(x)=v$.

Since $\psi _ x\leq 0$, dropping the factor $\mathrm e^{-\psi _ x}$ only decreases the integrals, and with $|G|_{h_0}^2\mathrm e^{-\varphi _ j}\leq |f|_ h^2$ on the ball we obtain

where $C_2$ depends only on the fixed geometric cutoffs and on the constant of the local point estimate.

Step 4: the limit. Since $\varphi _ j\leq \varphi _1\leq \sup _ X\varphi _1$, the sections $s_ j$ are bounded in $L^2$ for $h_0$, and by lemma 10.6.2(2) a subsequence converges uniformly to a holomorphic section $s$ of $L$, which satisfies $s(x)=v$. Since $\varphi _ j\downarrow \varphi $, Fatou's lemma gives (10.7.10) with $C=C_2$. $\square$