Lemma 6.12.115E8D84189EA2

Proof. We may assume $k<\infty $. Choose coordinates $w=(w_1,w')$ at $p$ with $E=\{ w_1=0\} $, and write $h=w_1^ kv$ with $v$ holomorphic and not divisible by $w_1$; then $v(0,\cdot )$ does not vanish identically near $p$, by lemma 6.12.1(2). Moving $p$ slightly along $E$, we may assume that $v(p)\neq 0$, so that $|v|\geq \delta >0$ on a neighborhood $\{ |w_1|<\rho \} \times W'$ of $p$ on which $|h|^2\mathrm e^{-\psi }$ is integrable. By Fubini's theorem, for almost every $w'\in W'$ the function $w_1\mapsto |w_1|^{2k}|v|^2\mathrm e^{-\psi (w_1,w')}$ is integrable on $|w_1|<\rho $; moreover, for almost every $w'$ the function $u=\psi (\cdot ,w')$ is not identically $-\infty $ on this disc, since the pole set of $\psi $ is a null set.

Fix such a $w'$. By proposition 5.9.3, $u$ is subharmonic on the disc, and its Lelong number at $0$ is at least $\nu (\psi ,(0,w'))$: restricting the bound $\psi (w)\leq c\log \bigl (|w-(0,w')|^2/r_0^2\bigr )+C$ of (6.7.3), with $c=\nu (\psi ,(0,w'))$, to the line through $(0,w')$ in the $w_1$-direction gives $u(w_1)\leq c\log (|w_1|^2/r_0^2)+C$. Moreover $\nu (\psi ,(0,w'))\geq \nu (\psi ,E)$, the latter being an infimum over $E$. Hence, by (6.7.3) for $u$, there are $r_1\in (0,\rho ]$ and $c'>0$ with $\mathrm e^{-u(w_1)}\geq c'|w_1|^{-2\nu (\psi ,E)}$ for $0<|w_1|<r_1$. Therefore

which forces $2k-2\nu (\psi ,E)+2>0$, that is, $k+1>\nu (\psi ,E)$. $\square$