6.8 Coherent ideals from weighted integrabilityC6A4B68C3F0B

Proof. We derive the intersection formula from the Artin–Rees lemma, after collecting the facts about Noetherian rings that we need. A commutative ring $A$ is Noetherian if all its ideals are finitely generated. Let $A$ be Noetherian.

(i) Every submodule $N$ of $A^ r$, $r\geq 1$, is finitely generated. For $r=1$ this is the definition. Let $r\geq 2$. The last coordinates of the elements of $N$ form an ideal of $A$; choose $v_1,\ldots ,v_ q\in N$ whose last coordinates generate it, and, by induction on $r$, finitely many generators of the submodule $N\cap (A^{r-1}\times \{ 0\} )$, which is a submodule of $A^{r-1}$. If $v\in N$, subtracting from $v$ a combination of the $v_ i$ with the same last coordinate leaves an element of $N\cap (A^{r-1}\times \{ 0\} )$; hence these finitely many elements generate $N$.

(ii) Every submodule $N$ of a finitely generated $A$-module $M$ is finitely generated. Indeed, let $\pi \colon A^ r\to M$ map the standard basis to generators of $M$. By (i), the submodule $\pi ^{-1}(N)$ of $A^ r$ is finitely generated, and the images of its generators generate $N=\pi (\pi ^{-1}(N))$.

(iii) The polynomial ring $A[t]$ is Noetherian (Hilbert's basis theorem). Let $I\subset A[t]$ be a nonzero ideal. The leading coefficients of the nonzero elements of $I$, together with $0$, form an ideal of $A$: if $f,g\in I$ have degrees $d\leq e$ and leading coefficients $a,b$ with $a+b\neq 0$, then $t^{e-d}f+g\in I$ has leading coefficient $a+b$, and if $c\in A$ with $ca\neq 0$, then $cf\in I$ has leading coefficient $ca$. Choose $p_1,\ldots ,p_ k\in I$ whose leading coefficients generate this ideal, and let $d_0$ be the largest of their degrees. If $f\in I$ has degree $d\geq d_0$, its leading coefficient is $\sum _ ia_ ic_ i$ for some $a_ i\in A$, where $c_ i$ is the leading coefficient of $p_ i$, and $f-\sum _ ia_ it^{d-\deg p_ i}p_ i$ is an element of $I$ of smaller degree. Repeating this, we see that $f$ differs from an element of $I\cap (A+At+\cdots +At^{d_0-1})$ by an element of the ideal generated by $p_1,\ldots ,p_ k$. The set $I\cap (A+At+\cdots +At^{d_0-1})$ is a submodule of the $A$-module $A+At+\cdots +At^{d_0-1}\cong A^{d_0}$, finitely generated by (i). Hence $I$ is generated by $p_1,\ldots ,p_ k$ and finitely many further elements.

(iv) If $I\subset A$ is an ideal, then $A/I$ is Noetherian, since the ideals of $A/I$ are the images of the ideals of $A$ containing $I$. Hence, by (iii) and induction on the number of variables, every quotient of a polynomial ring in finitely many variables over $A$ is Noetherian.

All this applies to $A=R$, which is Noetherian by proposition 9.2.4, since a holomorphic chart at $x$ identifies $R$ with $\mathcal O_ n$.

Next we prove the Artin–Rees lemma. If $\mathfrak m=0$, which happens only in dimension zero, the lemma is trivial; so let $\mathfrak m\neq 0$, and choose generators $\xi _1,\ldots ,\xi _ p$ of $\mathfrak m$. Consider the subring $\mathcal R=\bigoplus _{k\geq 0}\mathfrak m^ kt^ k$ of $R[t]$, consisting of the polynomials whose coefficient of $t^ k$ lies in $\mathfrak m^ k$, where $\mathfrak m^0=R$. Since $\mathfrak m^ k$ is generated by the monomials of degree $k$ in $\xi _1,\ldots ,\xi _ p$, the ring $\mathcal R$ is the image of the homomorphism $R[T_1,\ldots ,T_ p]\to R[t]$, $T_ i\mapsto \xi _ it$, and it is Noetherian by (iii) and (iv).

In the $\mathcal R$-module of polynomials in $t$ with coefficients in $F$, consider the submodule $\bigoplus _{k\geq 0}\mathfrak m^ kFt^ k$. It is generated over $\mathcal R$ by generators of $F$ placed in degree zero, and it contains the submodule $\mathcal E=\bigoplus _{k\geq 0}(E\cap \mathfrak m^ kF)t^ k$, because $\mathfrak m^ j(E\cap \mathfrak m^ kF)\subset E\cap \mathfrak m^{j+k}F$. By (ii), $\mathcal E$ is finitely generated. The homogeneous components of the elements of $\mathcal E$ again belong to $\mathcal E$, so the homogeneous components of finitely many generators generate $\mathcal E$; let these be $e_1t^{d_1},\ldots ,e_ lt^{d_ l}$ with $e_ i\in E\cap \mathfrak m^{d_ i}F$, and let $s=\max _ id_ i$. For $k\geq s$ this gives the Artin–Rees equality

Indeed, let $y\in E\cap \mathfrak m^ kF$, and write $yt^ k=\sum _ i\rho _ ie_ it^{d_ i}$ with $\rho _ i\in \mathcal R$. Comparing the coefficients of $t^ k$, we may replace each $\rho _ i$ by its term of degree $k-d_ i$, which is $a_ it^{k-d_ i}$ with $a_ i\in \mathfrak m^{k-d_ i}$. Thus $y=\sum _ ia_ ie_ i$, where $a_ i\in \mathfrak m^{k-d_ i}=\mathfrak m^{k-s}\mathfrak m^{s-d_ i}$ and $\mathfrak m^{s-d_ i}e_ i\subset E\cap \mathfrak m^ sF$; hence $y\in \mathfrak m^{k-s}(E\cap \mathfrak m^ sF)$. The reverse inclusion is immediate.

Now let $Q$ be a finitely generated $R$-module and $G=\bigcap _ k\mathfrak m^ kQ$. By (ii), $G$ is finitely generated. Let $s$ be the index of the Artin–Rees equality for $G\subset Q$. Since $G\subset \mathfrak m^ kQ$ for all $k$, this equality with $k=s+1$ gives $G=G\cap \mathfrak m^{s+1}Q=\mathfrak m(G\cap \mathfrak m^ sQ)=\mathfrak mG$. If $g_1,\ldots ,g_ r$ generate $G$, write $g_ i=\sum _ ja_{ij}g_ j$ with $a_{ij}\in \mathfrak m$. Multiplying this system of equations by the adjugate of the matrix $(\delta _{ij}-a_{ij})$ gives $\det (\delta _{ij}-a_{ij})\, g_ i=0$ for every $i$. Expanding the determinant, we see that it is congruent to $1$ modulo $\mathfrak m$, hence a unit of the local ring $R$, and therefore $G=0$. Applied to $Q=F/E$, for which $\mathfrak m^ kQ=(E+\mathfrak m^ kF)/E$, this proves the intersection formula.

For the last assertion, let $f_ l\to f$ locally uniformly near $x$, with germs $f_ l\in J$, and work in a holomorphic chart centered at $x$. By section 9.2, a germ belongs to $\mathfrak m^ k$ if and only if its Taylor coefficients of order less than $k$ vanish; hence these coefficients identify $R/\mathfrak m^ k$ with a finite-dimensional vector space. By the integral formula for the coefficients in (5.2.1), the Taylor coefficients of $f_ l$ at $x$ converge to those of $f$, so the images of the $f_ l$ in $R/\mathfrak m^ k$ converge to the image of $f$. The image of $J$ there is a linear subspace of a finite-dimensional vector space, hence closed, so $f\in J+\mathfrak m^ k$ for every $k$. The intersection formula with $F=R$ and $E=J$ gives $f\in J$. $\square$

Proof. The set of points $z$ with $\mathcal I_ z=0$ is open: near such a point, choose sections $g_1,\ldots ,g_ r$ generating all stalks of $\mathcal I$ on a neighborhood; their germs at $z$ vanish, so they vanish on a neighborhood $U'$ of $z$, and $\mathcal I$ has zero stalks on $U'$. The set of points with $\mathcal I_ z\neq 0$ is open as well: a section with nonzero germ at $z$, defined on a connected neighborhood of $z$, has nonzero germs at all points of this neighborhood, by the identity theorem. Since $W$ is connected, one of the two sets is empty. $\square$

Proof. For $z'\in W'$, the polynomial $P(z',\cdot )$ has at most $\mu $ roots, all in $\Delta $; so the fibers of $Z$ over $W'$ are finite. We first describe $Z$ near a fiber. Let $b\in W'$, let $w_1,\ldots ,w_ l$ be the distinct roots of $P(b,\cdot )$, and put $z_ i=(b,w_ i)$. Let $\delta >0$ be so small that the closed discs $|w-w_ i|\leq 2\delta $ are pairwise disjoint and contained in $\Delta $. The function $|P|$ is continuous and positive on the compact set $\{ b\} \times K$, where $K=\{ |w|\leq \rho \} \setminus \bigcup _ i\{ |w-w_ i|<\delta \} $; hence it is positive on $V_ b\times K$ for a neighborhood $V_ b$ of $b$, and for $z'\in V_ b$ all roots of $P(z',\cdot )$ lie in $\bigcup _ i\{ |w-w_ i|<\delta \} $. Put $B_ i=V_ b\times \{ |w-w_ i|<2\delta \} $. These open sets are pairwise disjoint, and $Z\cap (V_ b\times \Delta )\subset \bigcup _ iB_ i$. Moreover, given neighborhoods $N_ i$ of the points $z_ i$, we can achieve $V_ b\times \{ |w-w_ i|\leq 2\delta \} \subset N_ i$ for all $i$: first decrease $\delta $ so that $\{ b\} \times \{ |w-w_ i|\leq 2\delta \} \subset N_ i$, and then decrease $V_ b$, using the compactness of these discs. We call such $\delta $ and $V_ b$ adapted to $b$; a smaller neighborhood of $b$ remains adapted.

(1) Let $s$ be a section of $\mathcal G$ over $V'\times \Delta $ whose germs at all the $z_ i$ vanish. Then $s$ vanishes on neighborhoods $N_ i$ of the $z_ i$; choose $\delta $ and $V_ b\subset V'$ adapted to $b$ with $B_ i\subset N_ i$. The open sets $B_1,\ldots ,B_ l$ and $(V_ b\times \Delta )\setminus Z$ cover $V_ b\times \Delta $, and $s$ vanishes on each of them, on the last one because all stalks of $\mathcal G$ vanish there. Hence $s=0$ on $V_ b\times \Delta $, and the map is injective. Conversely, let $g_ i\in \mathcal G_{z_ i}$ be given, represented by sections $t_ i$ on neighborhoods $N_ i$ of the $z_ i$, and choose $\delta $ and $V_ b$ adapted to $b$ with $B_ i\subset N_ i$. On the open cover of $V_ b\times \Delta $ formed by the $B_ i$ and $(V_ b\times \Delta )\setminus Z$, consider the sections $t_ i|_{B_ i}$ and $0$. They agree on the overlaps: the $B_ i$ are pairwise disjoint, and $t_ i$ vanishes on $B_ i\setminus Z$, since $\mathcal G$ has zero stalks there. Gluing them, we obtain a section over $V_ b\times \Delta $ with germs $g_ i$ at the $z_ i$. Hence the map is surjective.

(2) The map $\Phi $ is a morphism of sheaves of $\mathcal O_{W'}$-modules. It is injective on sections: suppose that the class of $R=\sum _{k<\mu }c_ kw^ k$ vanishes on $V'\times \Delta $, and let $z'\in V'$. If $w_0$ is a root of $P(z',\cdot )$ of multiplicity $m$, then $R=Pq$ near $(z',w_0)$ with $q$ holomorphic, so the polynomial $R(z',\cdot )=P(z',\cdot )q(z',\cdot )$ vanishes to order at least $m$ at $w_0$. Thus $R(z',\cdot )$, a polynomial of degree less than $\mu $, has at least $\mu $ zeros counted with multiplicity, and therefore vanishes; that is, $c_ k(z')=0$ for all $k$. To prove surjectivity on stalks, let $s$ be a section of $\mathcal O_ D/P\mathcal O_ D$ over $V'\times \Delta $ and $b\in V'$. Near each $z_ i$, $s$ is the class of a holomorphic function $f_ i$ on a neighborhood $N_ i$ of $z_ i$. Choose $\delta $ and $V_ b\subset V'$ adapted to $b$ with $V_ b\times \{ |w-w_ i|\leq 2\delta \} \subset N_ i$. On the circles $|\zeta -w_ i|=2\delta $ the function $P(z',\zeta )$ does not vanish for $z'\in V_ b$, since these circles have distance at least $\delta $ from all roots. For $z'\in V_ b$ and $w\in \mathbb C$ put

The difference quotient is a polynomial in $w$ of degree $\mu -1$ whose coefficients are polynomials in $\zeta $ and in the $a_ k(z')$; hence $R$ is a polynomial in $w$ of degree less than $\mu $ whose coefficients are holomorphic on $V_ b$, by differentiation under the integral sign. For $z'\in V_ b$ and $|w-w_ i|<2\delta $ put

a holomorphic function on $B_ i$. Adding the two integrals, we obtain

since the term $j=i$ equals $f_ i(z',w)$ by Cauchy's formula, and the terms $j\neq i$ vanish by Cauchy's theorem, the point $w$ lying outside the closed disc $|\zeta -w_ j|\leq 2\delta $. Hence $f_ i\equiv R$ modulo $P$ on $B_ i$, so $s$ and the class of $R$ agree on each $B_ i$; on $(V_ b\times \Delta )\setminus Z$ both vanish, since $P$ is a unit there. Thus $s=\Phi (c)$ on $V_ b\times \Delta $, where $c$ is the tuple of coefficients of $R$. Being injective, and surjective on all stalks, $\Phi $ is an isomorphism.

(3) The sheaf $\mathcal M(\mathcal I)$ is a subsheaf of $\mathcal O_{W'}$-modules of $\mathcal O_{W'}^\mu $. Since $\mathcal O_{W'}$ is coherent (theorem 9.2.7), the last assertion of lemma 9.2.6 shows that it suffices to prove that $\mathcal M(\mathcal I)$ is locally finitely generated. Note that $\mathcal I/P\mathcal O_ D$ has zero stalks off $Z$, because $P$ is a unit there; so (1) applies to it as well as to $\mathcal O_ D/P\mathcal O_ D$.

Fix $b\in W'$, with $z_ i$ as above. Since $\mathcal I$ is coherent, there are neighborhoods $N_ i$ of $z_ i$ and sections $f_{i1},\ldots ,f_{i\kappa }$ of $\mathcal I$ over $N_ i$ generating all stalks of $\mathcal I$ on $N_ i$. Choose $\delta $ and $V_ b$ adapted to $b$ with $B_ i\subset N_ i$. For $1\leq i\leq l$, $1\leq \lambda \leq \kappa $ and $0\leq m<\mu $, let $\sigma _{i\lambda m}$ be the section of $\mathcal I/P\mathcal O_ D$ over $V_ b\times \Delta $ that equals the class of $w^ mf_{i\lambda }$ on $B_ i$ and $0$ on the other $B_ j$ and on $(V_ b\times \Delta )\setminus Z$. As in (1), these local sections glue, since the $B_ j$ are pairwise disjoint and the class of $w^ mf_{i\lambda }$ vanishes on $B_ i\setminus Z$, where $P$ is a unit.

We claim that the sections $\Phi ^{-1}(\sigma _{i\lambda m})$ of $\mathcal M(\mathcal I)$ over $V_ b$ generate all its stalks. Let $b'\in V_ b$, and let $y_1,\ldots ,y_ p$ be the points of $Z$ over $b'$; each of them lies in exactly one $B_ i$. By (1), every element of $\pi _*(\mathcal I/P\mathcal O_ D)_{b'}$ is a sum of elements $e$ whose germs vanish at all points of the fiber except one point $y_ q$, where the germ is some $g\in (\mathcal I/P\mathcal O_ D)_{y_ q}$. Let $y_ q\in B_ i$, and write $g$ as the class of $\sum _\lambda h_\lambda f_{i\lambda }$ with germs $h_\lambda $ at $y_ q$. By (1) for $\mathcal O_ D/P\mathcal O_ D$ and by (2), for each $\lambda $ there is $c^{(\lambda )}\in \mathcal O_{W',b'}^\mu $ such that $R_\lambda =\sum _ mc^{(\lambda )}_ mw^ m$ is congruent to $h_\lambda $ modulo $P$ at $y_ q$ and to $0$ modulo $P$ at the other points of the fiber. Then the element $\sum _{\lambda ,m}c^{(\lambda )}_ m\sigma _{i\lambda m}$ of $\pi _*(\mathcal I/P\mathcal O_ D)_{b'}$ has the germ $g$ at $y_ q$ and the germ $0$ at every other point of the fiber: at the other points in $B_ i$ its germ is the class of $\sum _\lambda R_\lambda f_{i\lambda }\in P\mathcal O_ D$, and at the points outside $B_ i$ it vanishes by construction. By the injectivity in (1), it equals $e$. This proves the claim, and $\mathcal M(\mathcal I)$ is coherent.

Finally, let $\mathcal M(\mathcal I)=\mathcal M(\mathcal I')$ on $V'$, and let $z\in V'\times \Delta $. If $z\notin Z$, then $\mathcal I_ z=\mathcal O_{D,z}=\mathcal I'_ z$, since $P$ is a unit at $z$. Let $z\in Z$, $b=\pi (z)$, and $f\in \mathcal I_ z$. By (1), there is an element of $\pi _*(\mathcal O_ D/P\mathcal O_ D)_ b$ whose germ at $z$ is the class of $f$ and whose germs at the other points of the fiber vanish. It belongs to $\pi _*(\mathcal I/P\mathcal O_ D)_ b=\Phi (\mathcal M(\mathcal I)_ b)=\Phi (\mathcal M(\mathcal I')_ b)=\pi _*(\mathcal I'/P\mathcal O_ D)_ b$, so its germ at $z$ belongs to $(\mathcal I'/P\mathcal O_ D)_ z$. Hence $f\in \mathcal I'_ z+P\mathcal O_{D,z}=\mathcal I'_ z$. By symmetry, $\mathcal I_ z=\mathcal I'_ z$. $\square$

Proof. The assertion on compact sets follows from the local one by covering a compact set with finitely many of the neighborhoods $U$ and taking the largest of the corresponding indices. To prove the local assertion at a point $x$, note that $\mathcal F$, being locally finitely generated, is near $x$ the image of a surjective morphism $\phi \colon \mathcal O^ r\to \mathcal F$ defined by finitely many generators. The inverse images $\phi ^{-1}(\mathcal F_ j)$ form an increasing sequence of subsheaves of $\mathcal O^ r$, and they are coherent by lemma 9.2.6, being the kernels of the morphisms $\mathcal O^ r\to \mathcal F/\mathcal F_ j$ between coherent sheaves; here $\mathcal O^ r$ is coherent by theorem 9.2.7. Since $\mathcal F_ j=\phi (\phi ^{-1}(\mathcal F_ j))$, the sequence $(\mathcal F_ j)$ is stationary wherever $(\phi ^{-1}(\mathcal F_ j))$ is.

We may therefore assume that $\mathcal F=\mathcal O^ r$, and we reduce to the case of ideals, $r=1$, by induction on $r$. Let $p\colon \mathcal O^ r\to \mathcal O$ be the projection to the last coordinate. The images $p(\mathcal F_ j)$ form an increasing sequence of coherent ideals, and the kernels $\mathcal F_ j\cap (\mathcal O^{r-1}\oplus 0)$ of $p|_{\mathcal F_ j}$ form an increasing sequence of coherent subsheaves of $\mathcal O^{r-1}$, by lemma 9.2.6. Suppose that both sequences are constant for $j\geq j_0$ on a neighborhood $U$ of $x$. If $\sigma $ is a germ of $\mathcal F_ j$ at a point of $U$, with $j\geq j_0$, choose a germ $\sigma '$ of $\mathcal F_{j_0}$ with $p(\sigma ')=p(\sigma )$; then $\sigma -\sigma '$ lies in the kernel for the index $j$, which equals the kernel for $j_0$, so $\sigma \in \mathcal F_{j_0}$. Thus $\mathcal F_ j=\mathcal F_{j_0}$ on $U$.

For ideals $\mathcal I_1\subset \mathcal I_2\subset \cdots $ we argue by induction on the dimension $n$ of $X$ at $x$; by the reduction just made, the induction hypothesis applies to coherent subsheaves of $\mathcal O^\mu $ in dimension $n-1$. In dimension zero, $x$ is an open point of $X$ and the only ideals of $\mathcal O_{X,x}=\mathbb C$ are $0$ and $\mathbb C$, so the increasing sequence $(\mathcal I_{j,x})$ is eventually constant, and $U=\{ x\} $ has the required property.

Let $n\geq 1$, and let $W$ be a connected coordinate neighborhood of $x$. If all stalks $\mathcal I_{j,x}$ vanish, then all $\mathcal I_ j$ vanish on $W$ by lemma 6.8.2, and there is nothing to prove. Otherwise some $\mathcal I_{j_1}$ has a nonzero germ $g$ at $x$. After a linear change of coordinates $z=(z',w)$ centered at $x$, in which $g$ is distinguished in $w$ (section 9.2), the Weierstrass preparation theorem (theorem 5.5.2) shows that $g$ is a unit times a Weierstrass polynomial $P(z',w)$ of some degree $\mu $, so that the germ of $P$ belongs to $\mathcal I_{j_1,x}$. If $\mu =0$, then $g$ is a unit, the section $1=g\cdot g^{-1}$ belongs to $\mathcal I_{j_1}$ near $x$, and $\mathcal I_ j$ is the unit ideal near $x$ for all $j\geq j_1$.

Let $\mu \geq 1$. By (9.2.1) the roots of $P(z',\cdot )$ tend to $0$ as $z'\to 0$, so there is a product $D=W'\times \Delta \subset W$ of a polydisc $W'\subset \mathbb C^{n-1}$ centered at $0$ and a disc $\Delta =\{ |w|<r\} $ such that the coefficients of $P$ are holomorphic on $W'$, $P$ is a section of $\mathcal I_{j_1}$ over $D$, and all roots of $P(z',\cdot )$, $z'\in W'$, lie in $\{ |w|\leq \rho \} $ for some $\rho <r$. For $j\geq j_1$ the ideals $\mathcal I_ j|_ D$ contain $P$, and by lemma 6.8.3(3) the sheaves $\mathcal M(\mathcal I_ j|_ D)$ form an increasing sequence of coherent subsheaves of $\mathcal O_{W'}^\mu $. By the induction hypothesis, they are constant for $j\geq j_0$ on a neighborhood $V'$ of $0$, for some $j_0\geq j_1$, and lemma 6.8.3(3) gives $\mathcal I_ j=\mathcal I_{j_0}$ on the neighborhood $V'\times \Delta $ of $x$ for all $j\geq j_0$. This proves the local assertion.

Finally, let $h_1,h_2,\ldots $ be holomorphic functions on $X$. The ideals $\mathcal J_ j$ generated by $h_1,\ldots ,h_ j$ are locally finitely generated subsheaves of $\mathcal O$, hence coherent by theorem 9.2.7, lemma 9.2.6, and the stalk at $y$ of the ideal $\mathcal J$ generated by all the $h_ l$ is the union of the stalks $\mathcal J_{j,y}$. By the local assertion, every point has a neighborhood $U$ and an index $j_0$ with $\mathcal J_ j=\mathcal J_{j_0}$ on $U$ for $j\geq j_0$; then $\mathcal J=\mathcal J_{j_0}$ on $U$. Hence $\mathcal J$ is coherent, and near each point it is generated by $h_1,\ldots ,h_{j_0}$. $\square$

Proof. (1) The function $F$ is smooth with $\Delta F=4\sum _ j\partial _{z_ j}\partial _{\bar z_ j}F=0$, by (5.2.9); so its real and imaginary parts are harmonic and have the ball mean value property (proposition 3.1.5). Thus $F(y)$ is the mean of $F$ over $B(y,\delta )$, and the Cauchy–Schwarz inequality gives (6.8.7).

(2) Homogeneous polynomials $P_ l$, $P_{l'}$ of different degrees are orthogonal in $L^2$ of the unit sphere: $P_ l(\mathrm e^{\mathrm i\theta }\omega )=\mathrm e^{\mathrm il\theta }P_ l(\omega )$, and surface measure is invariant under the orthogonal maps $\omega \mapsto \mathrm e^{\mathrm i\theta }\omega $ (subsection 3.1.1), so $\int _{S^{2n-1}}P_ l\overline{P_{l'}}\, \mathrm dS=\mathrm e^{\mathrm i(l-l')\theta }\int _{S^{2n-1}}P_ l\overline{P_{l'}}\, \mathrm dS$ for every $\theta $. For small $t>0$ the homogeneous expansion $F(x+t\omega )=\sum _ lt^ lP_ l(\omega )$ converges uniformly in $\omega \in S^{2n-1}$, by (5.2.1); integrating term by term, we obtain $\int _{S^{2n-1}}|F(x+t\omega )|^2\, \mathrm dS(\omega )=\sum _{l\geq k}t^{2l}\int _{S^{2n-1}}|P_ l|^2\, \mathrm dS$, which implies the first inequality. The last integral is positive, because $P_ k$ does not vanish identically on the sphere, by homogeneity, and nonempty relatively open subsets of the sphere have positive surface measure. $\square$

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. $\square$

Proof. We write $\mathcal H=\mathcal H_\varphi (\Omega ,\varrho )$ and $\| \cdot \| =\| \cdot \| _{\varphi ,\varrho }$. Choose balls $V_1\supset V_2\supset \cdots $ centered at $x$, with radii tending to zero, inside the neighborhood on which the $g_ i$ generate $\mathcal I(\varphi )$. For positive integers $j$ and $k$ let $S_{j,k}$ be the set of $F\in \mathcal H$ that can be written as $F=\sum _ ig_ ih_ i$ on $V_ j$ with $h_ i\in \mathcal O(V_ j)$ and $\sup _{V_ j}|h_ i|\leq k$. These sets cover $\mathcal H$: by theorem 6.8.9(3), the germ at $x$ of each $F\in \mathcal H$ lies in $\mathcal I(\varphi )_ x$, so $F=\sum _ ig_ ih_ i$ on some neighborhood of $x$ with holomorphic $h_ i$, and these coefficients are bounded on a smaller ball $V_ j$. We show that each $S_{j,k}$ is closed and then apply Baire's theorem.

Let $F_ l\in S_{j,k}$ converge to $F$ in $\mathcal H$, with coefficients $h_{i,l}$. Then $F_ l\to F$ locally uniformly on $\Omega $, by theorem 6.8.9(1). The coefficients are bounded by $k$ on $V_ j$. Let $K\subset V_ j$ be compact, and let $\varepsilon >0$ be such that the closed polydiscs of radius $2\varepsilon $ centered at points of $K$ lie in $V_ j$. For $y\in K$ and $y'\in B(y,\varepsilon )$, the closed polydisc of radius $\varepsilon $ centered at $y'$ then lies in $V_ j$, and the coefficient bound in (5.2.1) on this polydisc bounds all first partial derivatives of all $h_{i,l}$ at $y'$ by $k/\varepsilon $. So the $h_{i,l}$ are Lipschitz with a common constant on the balls of radius $\varepsilon $ centered at points of $K$, and they are equicontinuous on $K$. By proposition 2.7.4, applied on a compact exhaustion of $V_ j$, and a diagonal argument, a subsequence converges locally uniformly on $V_ j$ to continuous functions $h_ i$ with $|h_ i|\leq k$. Locally uniform convergence implies convergence in the sense of distributions, so $\bar\partial h_ i=0$, and the continuous functions $h_ i$ are holomorphic by lemma 6.4.10(1) (if $n=0$, every function on $V_ j$ is holomorphic). Passing to the limit, we obtain $F=\sum _ ig_ ih_ i$ on $V_ j$, that is, $F\in S_{j,k}$.

By Baire's theorem, a complete metric space is not a countable union of closed subsets with empty interior. (If all the sets had empty interior, we could choose closed balls of radii at most $2^{-l}$, each contained in the interior of the previous one and disjoint from the $l$th set, since the interior of a ball minus a closed set with empty interior is a nonempty open set; their centers would form a Cauchy sequence whose limit lies in every ball but in none of the sets.) Hence some $S_{j,k}$ contains a ball $\{ F:\| F-F_0\| <\delta \} $. If $\| F\| <\delta $, then $F_0$ and $F_0+F$ both lie in $S_{j,k}$, and subtracting their representations writes $F$ on $V_ j$ with coefficients bounded by $2k$. For $F\neq 0$ we apply this to $\delta F/(2\| F\| )$ and obtain the first assertion with $V=V_ j$ and $C=4k/\delta $; for $F=0$ we take zero coefficients.

For the second assertion, apply the first part with $\varrho =1$ to the weight $m\varphi $, and let $V$ and $C$ be the resulting neighborhood and constant. Let $a\in V$ with $\varphi (a)>-\infty $. By corollary 6.5.8, applied on the pseudoconvex set $\Omega $ to the plurisubharmonic function $m\varphi $, there is $F\in \mathcal H_{m\varphi }(\Omega )$ with $F(a)=1$ and $\| F\| _{m\varphi }^2\leq C_\Omega \mathrm e^{-m\varphi (a)}$, where $C_\Omega $ depends only on $n$ and the diameter of $\Omega $; if $n=0$, then $F=1$ has these properties with $C_\Omega =1$. Writing $F=\sum _ ig_ ih_ i$ on $V$ and using the Cauchy–Schwarz inequality, we obtain

Taking logarithms gives (6.8.11) at $a$ with $C_ m=m^{-1}\log (rC^2C_\Omega )$. At points where $\varphi =-\infty $ the left side is $-\infty $; by the last display, these points include all common zeros of the $g_ i$. $\square$