Lemma 6.15.309CBB2B6BA0F

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. $\square$