6.15 Local Bergman kernels and ideal generators9C2A6446C90A

Proof. We regard $\Omega $ as a complex manifold, with the single chart given by the identity map. Then $\psi $ is a weight on $\Omega $ in the sense of section 6.8, the set $\Omega $ satisfies the hypotheses of theorem 6.8.9, and $\mathcal H$ is the space $\mathcal H_\psi (\Omega ,\varrho )$ of that theorem, with the same inner product. Hence (1) is contained in theorem 6.8.9(1) and (2), and (2) follows from theorem 6.8.9(3), because an initial segment of an enumerated orthonormal basis is a finite family of elements of $\mathcal H$. Finally, (3) is the first assertion of lemma 6.8.10, applied to the weight $\psi $ on $\Omega $. $\square$

Proof. We give the proof for the measure $\varrho \, \mathrm dV$; Lebesgue measure is the case $\varrho =1$. By lemma 6.15.1, applied with $\psi =mu$, the space $H_ m$ is a Hilbert space with an orthonormal basis $(e_\ell )$, evaluation is uniformly bounded on compact subsets of $\Omega $, and (6.15.2) holds.

We show that the map $z\mapsto (e_\ell (z))_\ell $ into $\ell ^2$ is given locally by a convergent power series with coefficients in $\ell ^2$. Fix a point $z_0$ and a polydisc $P$ of multiradius $\rho $ about $z_0$ with $\overline P\subset \Omega $, and let $C$ be the evaluation bound on $\overline P$. For a finitely supported sequence $(c_\ell )$ with $\sum _\ell |c_\ell |^2\leq 1$, the function $F=\sum _\ell c_\ell e_\ell $ has norm at most one, so $|F|\leq C$ on $P$, and the Cauchy estimates give $|\partial ^\alpha F(z_0)/\alpha !|\leq C\rho ^{-\alpha }$. Since $\partial ^\alpha F(z_0)/\alpha !=\sum _\ell c_\ell A_{\alpha ,\ell }$ with $A_{\alpha ,\ell }=\partial ^\alpha e_\ell (z_0)/\alpha !$, taking the supremum over such sequences shows that the sequence $A_\alpha =(A_{\alpha ,\ell })_\ell $ has $\ell ^2$ norm $\| A_\alpha \| \leq C\rho ^{-\alpha }$. Let $0<\lambda <1$, and let $P_\lambda $ be the concentric polydisc of multiradius $\lambda \rho $. For $z\in P_\lambda $ we have $\sum _\alpha \| A_\alpha \| |(z-z_0)^\alpha |\leq C\sum _\alpha \lambda ^{|\alpha |}<\infty $, so the $\ell ^2$-valued series $\sum _\alpha A_\alpha (z-z_0)^\alpha $ converges absolutely, and its $\ell $th component is the Taylor series of $e_\ell $, which converges to $e_\ell (z)$. By (6.15.2),

where $|\langle A_\alpha ,A_\beta \rangle |\leq C^2\rho ^{-\alpha -\beta }$. Hence this double power series, and each series obtained from it by differentiating term by term, converge uniformly on $P_{\lambda '}$ for every $\lambda '<\lambda $, since a derivative of order $k$ only introduces factors bounded by a polynomial of degree $k$ in $|\alpha |+|\beta |$. Therefore $K_ m$ is real analytic, in particular smooth.

Next choose members $g_1,\ldots ,g_ s$ of $H_ m$ generating $\mathcal I(mu)$ near a given point, by lemma 6.15.1(2); by the remark following definition 6.14.5, any other finite system of generators gives a comparable sum of squares. By the definition of $K_ m$, $|g_ i|^2\leq \| g_ i\| _ m^2K_ m$, so $\sum _ i|g_ i|^2\leq \bigl (\sum _ i\| g_ i\| _ m^2\bigr )K_ m$. Conversely, lemma 6.15.1(3) gives, on a fixed smaller neighborhood, a representation $F=\sum _ ig_ ih_ i$ with $|h_ i|\leq C'$ for every $F$ in the unit ball of $H_ m$. The Cauchy–Schwarz inequality gives $|F|^2\leq sC'^2\sum _ i|g_ i|^2$, and hence $K_ m\leq sC'^2\sum _ i|g_ i|^2$ there. These constants may depend on $m$, $u$ and $\varrho $.

Let $\pi $ and $D$ be as in the statement, let $y\in Y$, let $V$ be a neighborhood of $\pi (y)$ on which the bounded representation of the previous paragraph holds, and let $s_ D$ be a local equation of $D$ on a neighborhood $W\subset \pi ^{-1}(V)$ of $y$. Since $\pi ^*\mathcal I(mu)$ is generated by $s_ D$ on $W$, we may write $g_ i\circ \pi =s_ Dk_ i$ and $s_ D=\sum _ il_ i\, (g_ i\circ \pi )$ with holomorphic $k_ i$ and $l_ i$, after shrinking $W$; since $s_ D$ does not vanish identically on any open set, $\sum _ il_ ik_ i=1$. For $F\in H_ m$, the representation $F=\sum _ ig_ ih_ i$ on $V$ gives $F\circ \pi =s_ D\sum _ i(h_ i\circ \pi )k_ i$ on $W$, so

Hence evaluation of $(F\circ \pi )/s_ D$ is bounded on each compact subset of $W$, uniformly for $F$ in the unit ball of $H_ m$. The power-series argument above therefore applies, in local coordinates on $Y$, to the holomorphic functions $(e_\ell \circ \pi )/s_ D$, and shows that $b_ m=\sum _\ell |(e_\ell \circ \pi )/s_ D|^2$ is smooth; by (6.15.2), $K_ m\circ \pi =|s_ D|^2b_ m$.

To see that $b_ m$ is strictly positive, let $F$ lie in the unit ball of $H_ m$, with coefficients $c_\ell =\langle F,e_\ell \rangle $ and partial sums $S_ N$. The estimate in the last display, applied to $F-S_ N$, shows that $(S_ N\circ \pi )/s_ D\to (F\circ \pi )/s_ D$ pointwise on $W$; hence $(F\circ \pi )/s_ D=\sum _\ell c_\ell (e_\ell \circ \pi )/s_ D$, and the Cauchy–Schwarz inequality gives $|(F\circ \pi )/s_ D|^2\leq b_ m$. Taking $F=g_ i/\| g_ i\| _ m$ for the indices with $g_ i\neq 0$, we obtain $b_ m\geq |k_ i|^2/\| g_ i\| _ m^2$, and the functions $k_ i$ have no common zero. $\square$

Proof. Let $\Omega _0$ be a connected component of $\Omega $, and choose $a\in \Omega _0$ with $u(a)>-\infty $. By corollary 6.5.8, applied on $\Omega $ to the weight $mu$, some element of $H_ m$ takes the value $1$ at $a$, so $H_ m$ contains a function that does not vanish identically on $\Omega _0$. For every $F$ in the unit ball of $H_ m$ that is not identically zero on $\Omega _0$, $\log |F|^2$ is plurisubharmonic on $\Omega _0$ by proposition 5.10.3, and $\log K_ m$ is the supremum of these functions on $\Omega _0$. Since $K_ m$ is continuous by lemma 6.15.3, this family is locally uniformly bounded above, and $\log K_ m$ is upper semicontinuous with values in $[-\infty ,\infty )$, so it equals its upper semicontinuous regularization. Hence theorem 5.10.2 shows that $u_ m$ is plurisubharmonic. The comparison in lemma 6.15.3 gives $u_ m=m^{-1}\log \sum _ i|g_ i|^2+O(1)$ locally, with generators $g_ i$ of $\mathcal I(mu)$; this is the asserted type.

For the lower bound in (6.15.5), let $u(z)>-\infty $; at points of the pole set of $u$ there is nothing to prove. By corollary 6.5.8, applied on $\Omega $ to the weight $mu$, there is $F\in H_ m$ with $F(z)=1$ and $\| F\| _ m^2\leq C_1\mathrm e^{-mu(z)}$, where $C_1$ depends only on $n$ and $L$. Hence $K_ m(z)\geq |F(z)|^2/\| F\| _ m^2\geq C_1^{-1}\mathrm e^{mu(z)}$. For the upper bound, let $\| F\| _ m\leq 1$. The mean-value inequality (6.8.7) of lemma 6.8.6 for $|F|^2$ on $B(z,r)$, where $|B(z,r)|=\pi ^ nr^{2n}/n!$, gives

with $C_2=n!/\pi ^ n$, and taking the supremum over $F$ proves the upper bound.

Now fix $z$ and $r$, and let $m\to \infty $ in (6.15.5): we obtain $u(z)\leq \varliminf _ mu_ m(z)\leq \varlimsup _ mu_ m(z)\leq \sup _{B(z,r)}u$. Letting $r\downarrow 0$ and using the upper semicontinuity of $u$ gives $u_ m(z)\to u(z)$, also at points of the pole set. On a compact set $K\subset \Omega $, a fixed $r<\operatorname {dist}(K,\partial \Omega )$ in the upper bound gives a uniform upper bound for all $u_ m$, while $u_ m\geq u-|\log C_1|$. Since $u\in L^1(K)$, dominated convergence gives $u_ m\to u$ in $L^1(K)$.

For the Lelong numbers, let $g_1,\ldots ,g_ s$ generate $\mathcal I(mu)$ near $z$. The stalk $\mathcal I(mu)_ z$ is not zero: by the first paragraph, applied to the component of $\Omega $ containing $z$, it contains the germ of an element of $H_ m$ that does not vanish identically on that component, and this germ is not zero by the identity theorem. Since $u_ m=m^{-1}\log \sum _ i|g_ i|^2+O(1)$ near $z$, lemma 6.14.7 gives $\nu (u_ m,z)=m^{-1}\operatorname {ord}_ z\mathcal I(mu)$, and (6.15.6) follows from lemma 6.9.1.

Finally, near each point $K_ m\leq C\sum _ i|g_ i|^2$ with $g_ i\in H_ m$, by lemma 6.15.3, and each $|g_ i|^2\mathrm e^{-mu}$ is integrable on $\Omega $. Hence $\exp (m(u_ m-u))=K_ m\mathrm e^{-mu}$ is locally integrable.

Now replace Lebesgue measure by $\varrho \, \mathrm dV$, and let $K^\varrho _ m$ be the corresponding kernel. The space of holomorphic functions is the same, and the two norms satisfy $c_\varrho \| F\| _ m^2\leq \| F\| _{m,\varrho }^2\leq C_\varrho \| F\| _ m^2$. If $\| F\| _ m\leq 1$, then $F/C_\varrho ^{1/2}$ lies in the unit ball for $\varrho $, and if $\| F\| _{m,\varrho }\leq 1$, then $c_\varrho ^{1/2}F$ lies in the unit ball for Lebesgue measure; hence

The proof of plurisubharmonicity applies verbatim to $m^{-1}\log K^\varrho _ m$, with the unit ball taken for $\varrho $ and the continuity of $K^\varrho _ m$ given by lemma 6.15.3. The lower and upper bounds follow from the last display, with the constants $C_\varrho C_1$ and $C_2/c_\varrho $. The convergence and the Lelong numbers then follow as before, since lemma 6.15.3 applies to $K^\varrho _ m$. Finally, $K^\varrho _ m\mathrm e^{-mu}\leq c_\varrho ^{-1}K_ m\mathrm e^{-mu}$ is locally integrable. $\square$

Proof. We write $\| \cdot \| _ m$ for the norm of $H_ m$. By the definition of $K_ m$, every $h\in H_ m$ satisfies $|h(\zeta )|^2\leq \| h\| _ m^2K_ m(\zeta )$ for all $\zeta \in \Omega $; if $\| h\| _ m=0$, then $h$ vanishes almost everywhere, hence identically.

Let $F\in H_{2m}$. We first extend $F$ from the diagonal to $\Omega \times \Omega $. By corollary 6.3.5, $\Omega \times \Omega $ is a bounded pseudoconvex open set in $\mathbb C^{2n}$, of diameter at most $\sqrt2L$, and the function $p(z,w)=mu(z)+mu(w)$ is plurisubharmonic on it (proposition 5.9.3, proposition 5.9.5). The diagonal $\Lambda =\{ (z,w)\in \mathbb C^{2n}:z=w\} $ is a complex linear subspace of codimension $n$, and the map $z\mapsto (z,z)$ multiplies Euclidean lengths by $\sqrt2$, so Lebesgue measure on $\Lambda $ corresponds to $2^ n$ times Lebesgue measure on $\mathbb C^ n$. On $\Lambda \cap (\Omega \times \Omega )$, identified with $\Omega $, the weight $p$ restricts to $2mu$, so

The extension statement of corollary 6.5.8, applied on $\Omega \times \Omega $ to the weight $p$ and the subspace $\Lambda $, gives $G\in \mathcal O(\Omega \times \Omega )$ with $G(z,z)=F(z)$ and $\int _{\Omega \times \Omega }|G|^2\mathrm e^{-p}\, \mathrm dV_{4n}\leq C^ n(\sqrt2L)^{2n}2^ nc_\varrho ^{-1}\| F\| _{2m}^2$, where $C$ is the constant of theorem 6.5.4. Hence

with $C_\Omega =4^ nC^ nL^{2n}C_\varrho ^2/c_\varrho $.

We now bound $G$ on the diagonal by $K_ m$. For $w\in \Omega $ put

By Tonelli's theorem, $N$ is measurable and $\int _\Omega N(w)\mathrm e^{-mu(w)}\varrho (w)\, \mathrm dV(w)=\| G\| ^2<\infty $; since $\mathrm e^{-mu}\varrho >0$ almost everywhere, $N(w)<\infty $ for almost every $w$. For such $w$, the function $G(\cdot ,w)$ belongs to $H_ m$ with $\| G(\cdot ,w)\| _ m^2=N(w)$, so $|G(z,w)|^2\leq N(w)K_ m(z)$ for every $z$. Fix $z\in \Omega $. Integrating this inequality in $w$, we obtain

so $G(z,\cdot )\in H_ m$ with $\| G(z,\cdot )\| _ m^2\leq K_ m(z)\| G\| ^2$, and therefore $|G(z,w)|^2\leq K_ m(z)K_ m(w)\| G\| ^2$ for every $w$. Taking $w=z$,

Taking the supremum over the unit ball of $H_{2m}$ proves the lemma. $\square$