10.7 Compact interpolation and global Bergman kernels9BE85E662C13

Proof. In linear coordinates at $x$ with $\mathrm dz_ j=\sqrt2\, \theta _ j$ we have $\sigma =\frac{\mathrm i}{2}\sum _ j\mathrm dz_ j\wedge \mathrm d\bar z_ j=\sum _ j\mathrm dx_ j\wedge \mathrm dy_ j$ at $x$, so the Riemannian metric of $\sigma $ at $x$ is Euclidean in the real coordinates, and $\langle \mathrm dx_ j+\mathrm i\, \mathrm dy_ j,\mathrm dx_ k\pm \mathrm i\, \mathrm dy_ k\rangle =(1\pm 1)\delta _{jk}$ shows that the covectors $\theta _ j,\bar\theta _ k$ form an orthonormal basis of the complexified cotangent space at $x$. Hence the products $\theta _ I\wedge \bar\theta _ J$ form an orthonormal basis of the complex covectors at $x$. Let $\iota _ j$ and $\bar\iota _ j$ be the adjoints of the exterior products with $\theta _ j$ and $\bar\theta _ j$; thus $\iota _ j$ maps $\theta _ j\wedge \theta _ K\wedge \bar\theta _ L$ to $\theta _ K\wedge \bar\theta _ L$ for $j\notin K$ and annihilates the basis covectors that do not contain $\theta _ j$, and similarly for $\bar\iota _ j$. Since $\sigma \wedge {}$ is $\mathrm i\sum _ j(\theta _ j\wedge )(\bar\theta _ j\wedge )$, its adjoint is $\Lambda _\sigma =-\mathrm i\sum _ j\bar\iota _ j\iota _ j$.

(1) The covector $\gamma \wedge v$ has bidegree $(n+1,2)$ and vanishes, so $[\gamma ,\Lambda _\sigma ]v=\gamma \wedge \Lambda _\sigma v$. We have $\iota _ j(e\wedge \bar\theta _ k)=(-1)^{j-1}\theta _1\wedge \cdots \wedge \widehat{\theta _ j}\wedge \cdots \wedge \theta _ n\wedge \bar\theta _ k$, and $\bar\iota _ j$ maps this to $(-1)^{j+n}\delta _{jk}\, \theta _1\wedge \cdots \wedge \widehat{\theta _ j}\wedge \cdots \wedge \theta _ n$, the factor $\bar\theta _ k$ standing in the $n$th place. Hence $\Lambda _\sigma v=-\mathrm i\sum _ j(-1)^{j+n}v_ j\, \theta _1\wedge \cdots \wedge \widehat{\theta _ j}\wedge \cdots \wedge \theta _ n$. In the product of the $j$th term with $\gamma $ only $\mathrm i\lambda _ j\theta _ j\wedge \bar\theta _ j$ contributes, and $\theta _ j\wedge \bar\theta _ j\wedge \theta _1\wedge \cdots \wedge \widehat{\theta _ j}\wedge \cdots \wedge \theta _ n=(-1)^{n-j}e\wedge \bar\theta _ j$; so $\gamma \wedge \Lambda _\sigma v=\sum _ j\lambda _ jv_ j\, e\wedge \bar\theta _ j$. Since the covectors $e\wedge \bar\theta _ j$ are orthonormal, this gives the formula for $\langle [\gamma ,\Lambda _\sigma ]v,v\rangle _{\sigma ,h}$, and the last assertion follows because the $\lambda _ j$ are nonnegative if $\gamma \geq 0$ and positive if $\gamma $ is positive definite.

(2) We may assume that $\alpha \neq 0$. Since the assertion does not involve $\gamma $, we may choose the $\sigma $-unitary coframe with $\theta _1=\alpha /|\alpha |_\sigma $, by completing this unit covector to an orthonormal basis of the $(1,0)$-covectors. Then $\mathrm i\alpha \wedge \bar\alpha =\mathrm i|\alpha |_\sigma ^2\theta _1\wedge \bar\theta _1$, so (1) gives $\langle [\mathrm i\alpha \wedge \bar\alpha ,\Lambda _\sigma ]v,v\rangle _{\sigma ,h}=|\alpha |_\sigma ^2|v_1|_ h^2$. Writing $u=fe$, we have $\bar\alpha \wedge u=(-1)^ n|\alpha |_\sigma f\, e\wedge \bar\theta _1$, hence $|\langle \bar\alpha \wedge u,v\rangle _{\sigma ,h}|\leq |\alpha |_\sigma |f|_ h|v_1|_ h$, and $|f|_ h=|u|_{\sigma ,h}$.

(3) Write $g=e\wedge \tau =\sum _ j\tau _ j\, e\wedge \bar\theta _ j$ with an $F$-valued $(0,1)$-covector $\tau =\sum _ j\tau _ j\bar\theta _ j$. By (1), $\langle [\gamma ,\Lambda _\sigma ]^{-1}g,g\rangle _{\sigma ,h}=\sum _ j|\tau _ j|_ h^2/\lambda _ j=|\tau |_\gamma ^2$, the squared norm of $\tau $ in the metric induced by $\gamma $ on $(0,1)$-covectors, which does not involve $\sigma $. Moreover $\mathrm dV_\sigma =\sigma ^ n/n!=c_ n\, e\wedge \bar e$ with $c_ n=\mathrm i^ n(-1)^{n(n-1)/2}$. Hence $\langle [\gamma ,\Lambda _\sigma ]^{-1}g,g\rangle _{\sigma ,h}\, \mathrm dV_\sigma =c_ n|\tau |_\gamma ^2\, e\wedge \bar e$. If $\kappa =\mu ^{-1}e$, $\mu \neq 0$, is another nonzero $(n,0)$-covector, then $g=\kappa \wedge \mu \tau $ and $c_ n|\mu \tau |_\gamma ^2\, \kappa \wedge \bar\kappa =c_ n|\tau |_\gamma ^2\, e\wedge \bar e$. Thus the density equals $c_ n|\tau '|_\gamma ^2\, \kappa \wedge \bar\kappa $ whenever $g=\kappa \wedge \tau '$, an expression that does not involve $\sigma $. In the same way, $|u|_{\sigma ,h}^2\, \mathrm dV_\sigma =c_ n|f|_ h^2\, \kappa \wedge \bar\kappa $ for $u=f\kappa $. If $\gamma \geq a\sigma $, all $\lambda _ j$ are at least $a$, and $\sum _ j|\tau _ j|_ h^2/\lambda _ j\leq a^{-1}\sum _ j|\tau _ j|_ h^2=a^{-1}|g|_{\sigma ,h}^2$. Finally, the eigenvalues of $\sigma '$ with respect to $\sigma '$ itself are all equal to one, so $[\sigma ',\Lambda _{\sigma '}]$ is the identity on $(n,1)$-covectors by (1), and the independence just proved gives

if $\sigma '\geq c\sigma $. $\square$

Proof. If $e'=ge$ is a second holomorphic frame, with $g$ holomorphic and nowhere zero, then its weight is $p'=p-\log |g|^2$, and for a form $u'$ we have $\partial (gu')-\partial p\wedge gu'=g(\partial u'-\partial p'\wedge u')$, since $\partial \log |g|^2=g^{-1}\partial g$; so $D'$ does not depend on the frame. In a frame we have $D'D''+D''D'=\partial \bar\partial p\wedge $, and $\mathrm i\partial \bar\partial p=2\pi c_1(F,h)$. We prove the Kähler identities at a point, in coordinates and a frame adapted to $\sigma $ and $h$, and deduce the Bochner–Kodaira formula from them.

Fix a point, choose holomorphic coordinates centered at it in which $\sigma =\mathrm i\sum _ j\mathrm dz_ j\wedge \mathrm d\bar z_ j$ at the point and the first derivatives of the coefficients of $\sigma $ vanish there (lemma 10.2.11, after replacing the coordinates $z$ by $z/\sqrt2$), and a frame with $\partial p=0$ at the point: replacing a frame $e$ by $e\exp (\sum _ j\partial _ jp(0)z_ j)$ replaces $p$ by $p-2\operatorname{Re}\sum _ j\partial _ jp(0)z_ j$. Let $\epsilon _ k,\bar\epsilon _ k$ denote exterior multiplication by $\mathrm dz_ k,\mathrm d\bar z_ k$, and $\iota _ k,\bar\iota _ k$ their pointwise adjoints, the contractions. At the point they satisfy $\iota _ j\epsilon _ k+\epsilon _ k\iota _ j=\delta _{jk}$ and $\bar\iota _ j\bar\epsilon _ k+\bar\epsilon _ k\bar\iota _ j=\delta _{jk}$, while operators of different types anticommute; moreover $\Lambda _\sigma =-\mathrm i\sum _ j\bar\iota _ j\iota _ j$, as the adjoint of $\mathrm i\sum _ j\epsilon _ j\bar\epsilon _ j$.

The formal adjoints of $\sum _ k\epsilon _ k\partial _ k$ and $\sum _ k\bar\epsilon _ k\partial _{\bar k}$ are $-\sum _ k\iota _ k\partial _{\bar k}$ and $-\sum _ k\bar\iota _ k\partial _ k$, with $\iota _ k,\bar\iota _ k$ computed for the metric at each point, plus zero-order terms whose coefficients are linear in the first derivatives of the coefficients of $\sigma $ and of $\mathrm e^{-p}$. Indeed, in the frame $\mathrm dz_ I\wedge \mathrm d\bar z_ J$ the operators $\epsilon _ k$ have constant matrices, and the pointwise inner product of $F$-valued forms times the density of $\mathrm dV_\sigma $ is $\overline w^{\mathsf T}Hu$ for a smooth positive definite Hermitian matrix $H$ whose entries are smooth functions of the coefficients of $\sigma $ times $\mathrm e^{-p}$. For a constant matrix $A$ and compactly supported smooth $u,w$, integration by parts and $\partial _ k\overline w=\overline{\partial _{\bar k}w}$ give $\int \overline w^{\mathsf T}HA\, \partial _ ku=-\int \overline{(\partial _{\bar k}w)}^{\mathsf T}HAu-\int \overline w^{\mathsf T}(\partial _ kH)Au$, so the formal adjoint of $A\partial _ k$ is $-A^\dagger \partial _{\bar k}-H^{-1}A^*(\partial _{\bar k}H)$, where $A^\dagger =H^{-1}A^*H$ is the pointwise adjoint of $A$ and $A^*$ the conjugate transpose; the same computation applies with $\partial _ k$ and $\partial _{\bar k}$ exchanged. Since these first derivatives, and $\partial p$, vanish at the point, there

where $\partial _ k,\partial _{\bar k}$ act on the coefficients. The anticommutation rules give $\bar\iota _ j\iota _ j\epsilon _ k-\epsilon _ k\bar\iota _ j\iota _ j=\delta _{jk}\bar\iota _ j$ and $\bar\iota _ j\iota _ j\bar\epsilon _ k-\bar\epsilon _ k\bar\iota _ j\iota _ j=-\delta _{jk}\iota _ j$, hence $[\Lambda _\sigma ,D']=-\mathrm i\sum _ k\bar\iota _ k\partial _ k$ and $[\Lambda _\sigma ,D'']=\mathrm i\sum _ k\iota _ k\partial _{\bar k}$ at the point, since the coefficients of $\Lambda _\sigma $ in the frame $\mathrm dz_ I\wedge \mathrm d\bar z_ J$ have vanishing first derivatives there. These are the Kähler identities

which therefore hold everywhere, the point being arbitrary. Substituting them into $\Delta ''=D''D''{}^*+D''{}^*D''$ and $\Delta '=D'D'{}^*+D'{}^*D'$, the mixed compositions $D''\Lambda _\sigma D'$ and $D'\Lambda _\sigma D''$ cancel in the difference, and the curvature identity gives the Bochner–Kodaira formula

that is, $\Delta ''-\Delta '=B_\sigma $. For a compactly supported smooth $(n,1)$-form $v$, the form $D'v$ vanishes for degree reasons, and integrating $\langle (\Delta ''-\Delta ')v,v\rangle _\sigma $ over $M$, using the definition of the formal adjoints, gives (10.7.3). $\square$

Proof. On a compact subset of a coordinate chart over which $F$ has a holomorphic frame, the pointwise norms of $\sigma $ and $h_ k$ are comparable with the Euclidean norms of the coefficients, and $\mathrm dV_\sigma $ with Lebesgue measure. Hence convergence in $L^2_{(n,q)}(\sigma ,h_ k)$ implies $L^2$ convergence of the coefficients on such sets, and therefore convergence in the sense of distributions; since $\bar\partial $ is continuous for distributional convergence, $T$ and $S$ are closed. Compactly supported smooth forms belong to $\operatorname{Dom}T$ and $\operatorname{Dom}S$, and they are dense in $L^2_{(n,0)}(\sigma ,h_0)$ and in $L^2_{(n,1)}(\sigma ,h_1)$: if $(\psi _ a)$ is a partition of unity with compact supports in such charts (proposition 5.2.6), then an $L^2$ form $u$ is the $L^2$ limit of the finite partial sums of $\sum _ a\psi _ au$, by dominated convergence, and each $\psi _ au$ is the $L^2$ limit of the forms whose coefficients are the mollifications of those of $\psi _ au$, by lemma 2.5.1. Thus $T$ and $S$ are densely defined, and $ST=0$ because $\bar\partial \bar\partial =0$ on distributions.

If $v$ is a compactly supported smooth $(n,1)$-form and $u\in \operatorname{Dom}T$, then $\langle Tu,v\rangle $ is the value of the distribution $\bar\partial u$ on a smooth compactly supported form built from $v$, and the definition of the distributional derivative gives $\langle Tu,v\rangle =\langle u,\delta v\rangle $; hence $v\in \operatorname{Dom}T^*$ and $T^*v=\delta v$. For an arbitrary $v\in \operatorname{Dom}T^*$, testing with compactly supported smooth $u$ shows in the same way that $T^*v=\delta v$ in the sense of distributions, and $Sv=\bar\partial v$ in the sense of distributions for $v\in \operatorname{Dom}S$ by definition.

For the density, multiplication by the cutoffs $\chi _ R$ first reduces the problem to forms with compact support. Let $v\in \operatorname{Dom}T^*\cap \operatorname{Dom}S$ and $\chi =\chi _ R$. For $u\in \operatorname{Dom}T$, the Leibniz rule for distributions gives $\chi u\in \operatorname{Dom}T$ and $T(\chi u)=\chi Tu+\bar\partial \chi \wedge u$, so that $\langle Tu,\chi v\rangle =\langle T(\chi u),v\rangle -\langle \bar\partial \chi \wedge u,v\rangle =\langle u,\chi T^*v-(\bar\partial \chi \wedge )^*v\rangle $, where $(\bar\partial \chi \wedge )^*$ is the adjoint of the bounded operator $\bar\partial \chi \wedge $ from $L^2_{(n,0)}(\sigma ,h_0)$ to $L^2_{(n,1)}(\sigma ,h_1)$. Hence $\chi v\in \operatorname{Dom}T^*$ and $T^*(\chi v)=\chi T^*v-(\bar\partial \chi \wedge )^*v$; and the Leibniz rule gives $\chi v\in \operatorname{Dom}S$ with $S(\chi v)=\chi Sv+\bar\partial \chi \wedge v$. Since $|\bar\partial \chi _ R|_\sigma \leq |\mathrm d\chi _ R|_\sigma \leq C/R$ and the ratios $h_1/h_0$ and $h_2/h_1$ are bounded, the error terms have norm at most $C'\| v\| /R$, and $\chi _ Rv\to v$, $\chi _ RT^*v\to T^*v$, $\chi _ RSv\to Sv$ by dominated convergence.

Second, a compactly supported form is approximated by local convolutions, after a finite partition of unity (proposition 5.2.6). For a first-order operator $P=\sum _ la_ l(x)\partial _ l+b(x)$ with smooth matrix coefficients and a mollifier $\rho _ t$, the commutator $[P,\rho _ t*\, \cdot \, ]$ applied to $v$ equals

On a fixed compact set, the mean-value bound $|a_ l(x)-a_ l(y)|\leq C|x-y|$ and the scaling of $\rho _ t$ bound the kernels of these integral operators by $k_ t(x-y)$, where $\int k_ t$ is bounded independently of $t$. An integral operator with a kernel bounded by $k(x-y)$, $k\geq 0$ integrable, has norm at most $\int k$ on $L^2$, since by the Cauchy–Schwarz inequality $|\int k(x-y)v(y)\, \mathrm dy|^2\leq \int k\cdot \int k(x-y)|v(y)|^2\, \mathrm dy$; so these operators are bounded on $L^2$ uniformly in $t$. For smooth $v$ the commutator tends to zero in $L^2$, and by the uniform bound and the density of smooth functions in $L^2$ it tends to zero for every $v\in L^2$.

Now let $v\in \operatorname{Dom}T^*\cap \operatorname{Dom}S$ have compact support, and let $(\psi _ a)$ be a finite partition of unity near $\operatorname{Supp}v$ by functions with compact supports in charts over which $F$ has frames. By the product rules above, with $\psi _ a$ in place of $\chi $, each $w=\psi _ av$ lies in $\operatorname{Dom}T^*\cap \operatorname{Dom}S$. In the chart and frame, $\delta $ and $\bar\partial $ are first-order operators with smooth matrix coefficients, and $T^*w=\delta w$ and $Sw=\bar\partial w$ lie in $L^2$. The forms $w_ t$ whose coefficients are the mollifications of those of $w$ are smooth with compact support, so $T^*w_ t=\delta w_ t=\rho _ t*(\delta w)+[\delta ,\rho _ t*\, \cdot \, ]w$ and $Sw_ t=\rho _ t*(\bar\partial w)+[\bar\partial ,\rho _ t*\, \cdot \, ]w$. By lemma 2.5.1 and the commutator bound, $w_ t\to w$ in the graph norm as $t\to 0$, which proves the density. $\square$

Proof. The set $A$ is a null set, since $\psi $ is locally integrable, and in particular $A\neq X$. Choose $b\geq 0$ with $\mathrm{dd}^{\mathrm{c}}\psi \geq -b\omega $, which is possible by lemma 6.14.4(4), and subtract a constant so that $\psi <-2b-2$; this is possible because $\psi $ is upper semicontinuous on the compact manifold $X$. Put $\rho =-\log (-\psi )$ on $X\setminus A$. The chain rule gives

Since $0<-1/\psi <1/(2b+2)$, the first term is at least $-\omega /2$, and since $\partial \rho =-\partial \psi /\psi $, the form $\sigma =\omega +\mathrm{dd}^{\mathrm{c}}\rho $ satisfies

It is a Kähler form on $X\setminus A$. The rank-one form $\frac{\mathrm i}{2\pi }\partial \rho \wedge \bar\partial \rho $ has, in a $\sigma $-unitary coframe, the single eigenvalue $|\partial \rho |_\sigma ^2/(2\pi )$, so the last inequality gives $|\partial \rho |_\sigma ^2\leq 2\pi $; hence $|\mathrm d\rho |_\sigma $ is bounded by a numerical constant.

The function $-\rho =\log (-\psi )$ tends to $+\infty $ at $A$, because $\psi $ is upper semicontinuous and equal to $-\infty $ on $A$, and its sublevel sets $\{ -\rho \leq t\} =\{ \psi \geq -\mathrm e^ t\} $ are closed in $X$, because $\psi $ is upper semicontinuous, and disjoint from $A$; so they are compact subsets of $X\setminus A$. Choose a smooth function $\chi $ on $\mathbb R$ equal to one on $(-\infty ,1]$ and to zero on $[2,\infty )$, with $0\leq \chi \leq 1$, and put $\chi _ R=\chi (-\rho /R)$. Then $\chi _ R$ has compact support in $X\setminus A$, equals one on $\{ -\rho \leq R\} $, and $|\mathrm d\chi _ R|_\sigma \leq \sup |\chi '|\, |\mathrm d\rho |_\sigma /R$. The same gradient bound shows that every path in $X\setminus A$ that eventually leaves every compact subset of $X\setminus A$ has infinite $\sigma $-length, since $-\rho $ tends to $+\infty $ along it, by the compactness of its sublevel sets.

For the last assertion, recall that a function $\varphi $ with gentle analytic singularities of type $(c,\mathfrak a)$ is smooth outside its pole set, where it equals $c$ times the logarithm of the smooth positive function $\exp (\varphi /c)$ (definition 6.14.12). Hence a finite sum as in the statement is quasi-plurisubharmonic and smooth outside the union of the pole sets of its terms, and this union is its pole set, since all terms are bounded above. $\square$

Proof. Let $B_\sigma =[2\pi c_1(F,h),\Lambda _\sigma ]$. Since $c_1(F,h)$ is positive definite, $B_\sigma $ is positive definite on $(n,1)$-forms by lemma 10.7.1(1). Let $T=D''\colon L^2_{(n,0)}\to L^2_{(n,1)}$ and $S=D''\colon L^2_{(n,1)}\to L^2_{(n,2)}$ be the operators with maximal distributional domains, for the norms defined by $\sigma $ and $h$. By lemma 10.7.4, applied with $h_0=h_1=h_2=h$, they are closed and densely defined, $ST=0$, and compactly supported smooth forms are dense in $\operatorname{Dom}T^*\cap \operatorname{Dom}S$ for the graph norm $\| v\| +\| T^*v\| +\| Sv\| $.

Let $v$ be a compactly supported smooth $(n,1)$-form. We integrate the pointwise Cauchy–Schwarz inequality $|\langle g,v\rangle _\sigma |^2\leq \langle B_\sigma ^{-1}g,g\rangle _\sigma \langle B_\sigma v,v\rangle _\sigma $ for the positive operator $B_\sigma $ and use (10.7.3), in which $D''{}^*v=T^*v$ (lemma 10.7.4) and $D''v=Sv$. This gives

By lemma 10.7.1(3), the density $\langle B_\sigma ^{-1}g,g\rangle _\sigma \, \mathrm dV_\sigma $ does not depend on $\sigma $. Computing it with $\omega $ in place of $\sigma $, and using that $2\pi c_1(F,h)\geq 2\pi a\omega $, so that $\langle [2\pi c_1(F,h),\Lambda _\omega ]^{-1}g,g\rangle _{\omega ,h}\leq (2\pi a)^{-1}|g|_{\omega ,h}^2$ by the same lemma, we find that the first integral is at most the right side of (10.7.7).

The form $g$ lies in $\ker S$, and in $L^2_{(n,1)}$ for $\sigma $ and $h$, because $\sigma \geq \omega /2$ gives $|g|_{\sigma ,h}^2\, \mathrm dV_\sigma \leq 2|g|_{\omega ,h}^2\, \mathrm dV$, again by lemma 10.7.1(3). Hence both sides of the displayed inequality converge along a graph-convergent sequence, and by the density of compactly supported smooth forms the inequality extends to all $v\in \operatorname{Dom}T^*\cap \operatorname{Dom}S$. Theorem 6.2.4 therefore gives $u\in \operatorname{Dom}T$ with $Tu=g$ and $\| u\| _\sigma ^2$ at most the first integral. Finally, $|u|_{\sigma ,h}^2\, \mathrm dV_\sigma =|u|_{\omega ,h}^2\, \mathrm dV$ by lemma 10.7.1(3), which gives exactly (10.7.7). $\square$

Proof. The question is local. We show that the given function $f$ satisfies $\bar\partial f=0$ in the sense of distributions near each point of $A$, by a one-variable argument on complex lines. Near a point of $A$, the set $A$ is the common zero set of finitely many holomorphic functions on a connected coordinate neighborhood, and one of them, $b$, does not vanish identically, since otherwise $A$ would contain a nonempty open set. The lowest-order nonzero homogeneous term $P$ of $b$ at the point is a nonzero polynomial, so its zero set is closed and nowhere dense; hence we may choose linearly independent vectors $v_1,\ldots ,v_ n$ with $P(v_ k)\neq 0$, choosing each $v_ k$ outside the union of this zero set and the span of $v_1,\ldots ,v_{k-1}$. We use linear coordinates centered at the point in which the $v_ k$ are the coordinate unit vectors. Then $t\mapsto b(tv_ k)$ has a zero of finite order at $t=0$, so by the Weierstrass preparation theorem (theorem 5.5.2), on a sufficiently small polydisc, $b$ is a unit times a monic polynomial in the $k$th coordinate with holomorphic coefficients in the others, for each $k$. Hence every complex line parallel to a coordinate axis meets the zero set of $b$ in this polydisc in finitely many points.

By Fubini's theorem, $f$ is square integrable on almost every such line. On such a line, near each of the finitely many punctures, the Laurent expansion of $f$ has no terms of negative order: by orthogonality of the powers on circles, a term of order $-j$ with $j\geq 1$ would make the integral of $|f|^2$ diverge like $\int _0^ rt^{1-2j}\, \mathrm dt$. Hence the restriction of $f$ to almost every line extends holomorphically, and its $\bar\partial $-derivative along the line vanishes as a distribution. Integrating against compactly supported test functions and using Fubini's theorem, we find that every derivative $\partial f/\partial \bar z_ k$ vanishes as a distribution on the polydisc. A distribution with $\bar\partial f=0$ satisfies $\Delta f=4\sum _ k\partial _{z_ k}\partial _{\bar z_ k}f=0$, so it is smooth by Weyl's lemma (theorem 3.1.11) and therefore holomorphic. This holomorphic function agrees with $f$ on the complement of $A$, and the local extensions agree on overlaps. For sections of a line bundle, apply the result in local holomorphic frames. $\square$

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 $C=C_2$. $\square$

Proof. By lemma 10.6.2(3), $H^0(X,L^ k\otimes T)$ and its linear subspace $\mathcal H_ k$ are finite dimensional. The weighted integral defines an inner product on $\mathcal H_ k$: the function $\varphi $ is finite almost everywhere, so the weight is positive almost everywhere, and a section of norm zero vanishes almost everywhere and hence identically. A finite-dimensional inner product space is complete.

At a point $y$, with the coefficients $c_ i$ of $s=\sum _ ic_ is_ i$, the Cauchy–Schwarz inequality gives $|s(y)|^2\leq \sum _ i|c_ i|^2\sum _ i|s_ i(y)|^2$, with equality when the $c_ i$ are proportional to the complex conjugates of the values of the $s_ i$ at $y$ in a local frame. This proves the equality in (10.7.12). In local frames with $|e|_{h_0}^2=\mathrm e^{-p_0}$ and $|e_ T|_{h_ T}^2=\mathrm e^{-p_ T}$, write $s_ i=f_ ie^ k\otimes e_ T$. Then

and $\theta +k^{-1}\theta _ T+\mathrm{dd}^{\mathrm{c}}\varphi _ k=k^{-1}\mathrm{dd}^{\mathrm{c}}\log \sum _ i|f_ i|^2\geq 0$ by proposition 5.10.3. Hence $\varphi _ k\in \mathrm{PSH}(X,\theta +k^{-1}\theta _ T)$, with analytic singularities of type $(1/k,\mathfrak b)$, where $\mathfrak b$ is the ideal sheaf generated locally by the $f_ i$. This ideal sheaf does not depend on the frames, because a change of frames multiplies all $f_ i$ by the same nowhere vanishing holomorphic function, and it is coherent by lemma 6.8.4. Moreover $\mathrm e^{k\varphi _ k}=\sum _ i|s_ i|^2$ is smooth on $X$. The local expressions for $\varphi _ k$ above are presentations with the smooth remainders $-p_0-p_ T/k$, so proposition 6.14.8 shows that on every log resolution $\pi $ of $\mathfrak b$, with $\pi ^*\mathfrak b=\mathcal O(-D)$, locally $\varphi _ k\circ \pi $ is $k^{-1}\log |s_ D|^2$ plus a smooth function. These are the conditions of definition 6.14.12.

Apart from this appeal to definition 6.14.12, the proof uses only that $X$ is a compact complex manifold of positive dimension, since the last assertion of lemma 10.6.2 gives its part (3) in this generality; hence it also proves the last assertion of the lemma. $\square$