Theorem 6.8.9FE623FA661C4

Proof. If $n=0$, then $\Omega $ is empty, and there is nothing to prove, or a point $x$; in the latter case $\varphi (x)$ is finite, $\mathcal H_\varphi (\Omega ,\varrho )=\mathbb C$, and $\mathcal I(\varphi )_ x=\mathbb C$ is generated by every nonzero element, so all assertions are immediate. Assume therefore $n\geq 1$, and write $\mathcal H$, $\| \cdot \| $ and $\langle \cdot ,\cdot \rangle $ for $\mathcal H_\varphi (\Omega ,\varrho )$, its norm and its inner product. In the chart, $\varphi \in \mathrm{PSH}(\Omega )$ (section 5.9). We prove (1) and (2) by Hilbert space arguments, and (3) by solving a $\bar\partial $-equation with a weight that has a logarithmic pole.

We first prove (1). Let $K\subset \Omega $ be compact, let $r>0$ be half the distance from $K$ to $\mathbb C^ n\setminus \Omega $, and let $M$ be the maximum of the upper semicontinuous function $\varphi $ on the compact set $K_ r=\{ z:\operatorname {dist}(z,K)\leq r\} \subset \Omega $. Since $\mathrm e^{-\varphi }\varrho \geq c_\varrho \mathrm e^{-M}$ on $K_ r$, the inequality (6.8.7) on $\overline{B(z,r)}\subset K_ r$ gives, for $z\in K$ and $F\in \mathcal H$,

This proves (1).

Next we show that $\mathcal H$ is a separable Hilbert space. The measurable function $\varphi -\log \varrho $ is bounded above on compact subsets of $\Omega $, and $L^2(\Omega ,\mathrm e^{-\varphi +\log \varrho })$ is the $L^2$ space of the measure $\mathrm e^{-\varphi }\varrho \, \mathrm dV$. By lemma 6.4.10(2), applied to this function, the elements of this space that are represented by holomorphic functions form a closed subspace. A holomorphic function is determined by its class, since continuous functions that agree almost everywhere agree everywhere; hence $\mathcal H$ is identified with this closed subspace and is a Hilbert space.

Multiplication by $\mathrm e^{-\varphi /2}\varrho ^{1/2}$ maps $L^2(\Omega ,\mathrm e^{-\varphi +\log \varrho })$ isometrically into $L^2(\Omega )$, since $\varphi $ is locally integrable (theorem 5.9.1) and hence finite almost everywhere. The space $L^2(\Omega )$ is separable: by lemma 2.5.1(5), $C_ c(\Omega )$ is dense in it. Let $\mathcal S$ be the countable set of functions that take constant values in $\mathbb Q+\mathrm i\mathbb Q$ on finitely many half-open cubes $\prod _{i=1}^{2n}[m_ i2^{-k},(m_ i+1)2^{-k})$, $m_ i\in \mathbb Z$, $k\geq 1$, whose closures lie in $\Omega $, and vanish elsewhere. Let $g\in C_ c(\Omega )$. For large $k$, the cubes of side $2^{-k}$ of this form that meet $\operatorname{Supp}g$ lie in a fixed compact subset $K'$ of $\Omega $. Assigning to each of them a value in $\mathbb Q+\mathrm i\mathbb Q$ within $2^{-k}$ of the value of $g$ at one of its points, we obtain $s_ k\in \mathcal S$ vanishing outside $K'$ with $\sup |g-s_ k|\leq 2^{-k}+\sup \{ |g(y)-g(y')|:|y-y'|\leq \sqrt{2n}\, 2^{-k}\} $, which tends to zero by uniform continuity of $g$; hence $s_ k\to g$ in $L^2(\Omega )$, and $\mathcal S$ is dense in $L^2(\Omega )$. A subset of a separable metric space is separable, so $\mathcal H$ is separable.

Orthonormal bases of $\mathcal H$ exist: applying the Gram–Schmidt process to a countable dense sequence and discarding the vectors that depend linearly on the preceding ones, we obtain an orthonormal family whose linear span contains the dense sequence. Every orthonormal family is finite or countable, because two of its members have distance $\sqrt2$, so that the balls of radius $1/2$ around them are disjoint, and each of these balls contains a member of a fixed countable dense set. Let $(e_ j)_{j\geq 1}$ be an enumeration of an orthonormal basis, let $F\in \mathcal H$, and put $c_ j=\langle F,e_ j\rangle $ and $S_ N=\sum _{j\leq N}c_ je_ j$. Since $F-S_ N$ is orthogonal to the span $V_ N$ of $e_1,\ldots ,e_ N$, every $v\in V_ N$ satisfies $\| F-v\| ^2=\| F-S_ N\| ^2+\| S_ N-v\| ^2\geq \| F-S_ N\| ^2$. Given $\varepsilon >0$, there is $v$ in some $V_{N_0}$ with $\| F-v\| <\varepsilon $, and since $V_{N_0}\subset V_ N$ for $N\geq N_0$, we get $\| F-S_ N\| <\varepsilon $ for all $N\geq N_0$. Thus $S_ N\to F$ in $\mathcal H$, and hence locally uniformly on $\Omega $, by (1). Taking $v=0$ above gives $\| F\| ^2=\| F-S_ N\| ^2+\sum _{j\leq N}|c_ j|^2$, and letting $N\to \infty $ we obtain $\sum _ j|c_ j|^2=\| F\| ^2$. This proves (2).

We turn to (3). Let $\mathcal J$ be the ideal sheaf on $\Omega $ generated by the $e_ j$. By lemma 6.8.4 it is coherent and near each point generated by an initial segment $e_1,\ldots ,e_ k$. Every $F\in \mathcal H$ is the locally uniform limit of the partial sums $S_ N$, whose germs lie in $\mathcal J$, so the germs of $F$ lie in $\mathcal J$ by lemma 6.8.1. Thus $\mathcal J$ is the ideal generated by $\mathcal H$, independently of the basis, and $\mathcal J\subset \mathcal I(\varphi )|_\Omega $ because $\int _\Omega |F|^2\mathrm e^{-\varphi }\, \mathrm dV\leq c_\varrho ^{-1}\| F\| ^2$ for $F\in \mathcal H$.

It remains to prove that $\mathcal I(\varphi )_ x\subset \mathcal J_ x$ for $x\in \Omega $. Fix $f\in \mathcal I(\varphi )_ x$ and an integer $s\geq 0$; we construct $F\in \mathcal H$ with $f-F\in \mathfrak m_ x^{s+1}$, where $\mathfrak m_ x$ is the maximal ideal of $\mathcal O_{X,x}$. Choose a cutoff function $\chi \in C^\infty _ c(\Omega )$, equal to one near $x$, such that $f$ is holomorphic and $|f|^2\mathrm e^{-\varphi }$ is integrable on a neighborhood of $\operatorname{Supp}\chi $. The $(0,1)$-form $\alpha =f\bar\partial \chi $, extended by zero, is smooth, $\bar\partial $-closed and vanishes near $x$.

We apply theorem 6.4.9 on the pseudoconvex set $\Omega $ with the plurisubharmonic function $p=\varphi +(n+s)\log |z-x|^2$ (proposition 5.10.3, proposition 5.9.5) and the smooth function $q=|z|^2$, whose complex Hessian is the identity matrix. Since $|z-x|^{-2(n+s)}$ and $|\bar\partial \chi |$ are bounded on the support of $\alpha $, we have $\int _\Omega |\alpha |^2\mathrm e^{-p-q}\, \mathrm dV\leq C\int _{\operatorname{Supp}\chi }|f|^2\mathrm e^{-\varphi }\, \mathrm dV<\infty $. Hence there is $u$ with $\bar\partial u=\alpha $ and

Since $\mathrm e^{-p-q}$ is bounded below by a positive constant on compact sets, $u$ is locally square integrable. Near $x$ we have $\bar\partial u=0$, so, by lemma 6.4.10(1), $u$ agrees almost everywhere near $x$ with a holomorphic function; we replace $u$ by this function near $x$. We claim that $u$ vanishes to order at least $s+1$ at $x$. Since $\varphi $ is bounded above near $x$, the last display gives $\int _{B(x,r)}|u|^2|z-x|^{-2(n+s)}\, \mathrm dV<\infty $ for small $r>0$. We may assume that the germ of $u$ at $x$ is not zero, and we let $P_{k_0}\neq 0$ be the first nonzero homogeneous term of its expansion at $x$. Applying (6.8.8) of lemma 6.8.6 to $u$ and integrating in polar coordinates (3.1.2), we obtain

with $c>0$. This is finite only if $k_0\geq s+1$, which proves the claim.

Since $\Omega $ is bounded, $\mathrm e^{-|z|^2}|z-x|^{-2(n+s)}$ is bounded below on $\Omega $ by a positive constant, so $|u|^2\mathrm e^{-\varphi }$ is integrable on $\Omega $. The function $\chi f-u$ satisfies $\bar\partial (\chi f-u)=\alpha -\alpha =0$, so it agrees almost everywhere with a holomorphic function $F$ on $\Omega $, by lemma 6.4.10(1). Since $|F|^2\leq 2|\chi f|^2+2|u|^2$ almost everywhere, $|\chi f|^2\mathrm e^{-\varphi }$ is integrable by the choice of $\chi $, and $\varrho \leq C_\varrho $, we have $F\in \mathcal H$. Near $x$ the continuous functions $f-F$ and $u$ agree almost everywhere, hence everywhere, so $f-F=u\in \mathfrak m_ x^{s+1}$ by section 9.2. Thus $f\in \mathcal J_ x+\mathfrak m_ x^{s+1}$ for every $s$, and lemma 6.8.1 gives $f\in \mathcal J_ x$. Hence $\mathcal I(\varphi )|_\Omega =\mathcal J$, which proves (3).

Finally, every point of $X$ has a neighborhood $\Omega $ of the kind considered, namely a bounded polydisc in a chart. If $n\geq 1$, such a polydisc is pseudoconvex by corollary 6.3.5, being a product of discs, each of which is a sublevel set of the smooth plurisubharmonic exhaustion function $|\zeta -\zeta _0|^2$ of $\mathbb C$, where $\zeta _0$ is its center. By (3), $\mathcal I(\varphi )$ is generated near each point by finitely many holomorphic functions, so it is coherent by theorem 9.2.7 and the last assertion of lemma 9.2.6. $\square$