Corollary 6.12.14AF618D7A70E2

Proof. Let $k=\operatorname {ord}_ E\mathcal I(m\varphi )$, and choose a point $p\in E_{\mathrm{reg}}$. Put $y=\pi (p)$, choose a bounded polydisc $\Omega $ around $y$ in a chart on which $\varphi $ is bounded above, and choose $g_1,\ldots ,g_ r\in \mathcal H_{m\varphi }(\Omega )$ generating $\mathcal I(m\varphi )$ near $y$ (theorem 6.8.9(3)). Near $p$, in coordinates $w=(w_1,w')$ with $E=\{ w_1=0\} $, let $k_ i$ be the order of $g_ i\circ \pi $ along $E$ at $p$, so that $k=\min _ ik_ i$; since $\mathcal I(m\varphi )_ y\neq 0$, this minimum is finite.

For the lower bound, choose $i$ with $k_ i=k$. Since $|g_ i|^2\mathrm e^{-m\varphi }$ is integrable on $\Omega $, proposition 6.12.4, applied to the weight $m\varphi $, shows that $|h|^2\mathrm e^{-m\varphi \circ \pi }$, with $h=(g_ i\circ \pi )J_\pi $, is integrable near $p$. The order of $h$ along $E$ at $p$ is $k+A_ X(E)-1$, because orders add under products. By lemma 6.12.11, applied to the weight $m\varphi \circ \pi $, whose generic Lelong number along $E$ is $m\nu (\varphi ,E)$ by proposition 6.7.9, we obtain $k+A_ X(E)>m\nu (\varphi ,E)$. This is the left inequality.

For the upper bound, lemma 6.8.10 gives $m\varphi \leq \log \sum _ i|g_ i|^2+mC_ m$ on a neighborhood of $y$. Near $p$ we have $g_ i\circ \pi =w_1^{k_ i}u_ i$ with $u_ i$ holomorphic, so $|g_ i\circ \pi |\leq C|w_1|^{k_ i}\leq C|w_1|^ k$ where $|w_1|\leq 1$; it follows that $m\varphi \circ \pi \leq k\log |w_1|^2+C''$ on a neighborhood $N$ of $p$. For $q\in E\cap N$ we have $|w_1|\leq |w-q|$, so $m\varphi \circ \pi (w)\leq k\log |w-q|^2+C''$ near $q$, and proposition 6.7.1, proposition 6.7.9 give $\nu (\varphi \circ \pi ,q)\geq k/m$. Since the generic value $\nu (\varphi \circ \pi ,E)$ is attained at points of $E$ arbitrarily close to $p$ (proposition 6.9.7), we conclude that $m\nu (\varphi ,E)\geq k$. Dividing (6.12.15) by $m$ and letting $m\to \infty $ gives the limit formula. $\square$