Lemma 6.17.3C32B34A0713B

Proof. We first bound $G_ m$ from below. Let $x\in U_ i$. Then $R_ m(x)\geq a_{i,m}(x)$, and the lower bound in (6.15.5), in the version of theorem 6.15.4 with the measure $\omega ^ n/n!$, gives $m^{-1}\log K_{i,m}(x)\geq \varphi (x)+h_ i(x)-C/m$. Since $h_ i$ cancels, $G_ m\geq v_{i,m}\geq \varphi -C/m$ on $U_ i$, where $C\geq 0$ depends only on the fixed geometry.

For an upper bound, let $r>0$ be smaller than the distance, in the coordinates of $B_ j$, from $\operatorname{Supp}\chi _ j$ to $\partial B_ j$ for every $j$, and for $x\in \operatorname{Supp}\chi _ j$ let $B_ j(x,r)\subset B_ j$ denote the ball of radius $r$ about $x$ in these coordinates. If $\chi _ j(x)\neq 0$, the upper bound in (6.15.5) on $B_ j$, again in the version with the measure $\omega ^ n/n!$, gives

where $C_2$ depends only on the fixed geometry. By the mean value theorem, $\sup _{B_ j(x,r)}h_ j-h_ j(x)\leq C_ hr$, where $C_ h$ bounds the gradients of the finitely many functions $h_ j$ on the compact sets $\overline{B_ j}$ and therefore depends on $\theta $. Since $R_ m(x)$ is a sum of at most $J$ terms, each at most $\max _ ja_{j,m}(x)$ over the indices with $\chi _ j(x)\neq 0$,

Fix $x$. Letting first $m\to \infty $ and then $r\downarrow 0$, and using that the balls $B_ j(x,r)$ shrink to $x$, the upper semicontinuity of $\varphi $ and the lower bound, we obtain $G_ m(x)\to \varphi (x)$, also at points of the pole set. For a fixed $r$, the upper bound shows that $G_ m\leq \sup _ X\varphi +C_ hr+|\log (JC_2r^{-2n})|$ on $X$ for every $m$, where $\sup _ X\varphi <\infty $ because $\varphi $ is upper semicontinuous on the compact space $X$. Since moreover $\varphi -C\leq G_ m$ and $\varphi \in L^1(X)$, dominated convergence gives $G_ m\to \varphi $ in $L^1(X)$.

By lemma 6.17.1(1), $G_ m=m^{-1}\log \sum _ k|g_ k|^2+O(1)$ near $x$ with generators $g_ k$ of $\mathcal I(m\varphi )$. Since $G_ m\geq \varphi -C/m$ and $\varphi $ is locally integrable (lemma 6.14.4(1)), $G_ m$ is not identically $-\infty $ near $x$, so the germ of some $g_ k$ at $x$ is not zero and $\mathcal I(m\varphi )_ x\neq 0$. Hence lemma 6.14.7 gives $\nu (G_ m,x)=m^{-1}\operatorname {ord}_ x\mathcal I(m\varphi )$. Write $\varphi =u+f$ on a connected neighborhood of $x$, with $u$ plurisubharmonic and $f$ smooth; then $\mathcal I(m\varphi )=\mathcal I(mu)$ and $\nu (\varphi ,x)=\nu (u,x)$ near $x$, and the Lelong bounds follow from lemma 6.9.1, applied to the weight $u$.

For the integrability, note that $\mathrm e^{m(G_ m-\varphi )}=R_ m\mathrm e^{-m\varphi }=\sum _ j\chi _ j^2K_{j,m}\mathrm e^{-m(\varphi +h_ j)}$. Cover the compact set $\operatorname{Supp}\chi _ j$ by finitely many open sets on which $K_{j,m}\leq C_ m\sum _ i|g_ i|^2$ with members $g_ i$ of the weighted space on $B_ j$, by lemma 6.15.3. Since these $g_ i$ have finite weighted norm, every summand is integrable.

For the doubling inequality, lemma 6.15.7, applied on each $B_ j$ with $u=\varphi +h_ j$ and the measure $\omega ^ n/n!$, gives $K_{j,2m}\leq C_*K_{j,m}^2$, hence $a_{j,2m}\leq C_*a_{j,m}^2$, with $C_*$ depending only on the fixed geometry. Let $x\in U_ i$. For every $j$ with $\chi _ j(x)\neq 0$, lemma 6.16.2 gives $a_{j,m}(x)\leq \mathrm e^{C_ B}a_{i,m}(x)$, and $R_ m(x)\geq a_{i,m}(x)$. Consequently

at $x$, also when both sides vanish. Taking logarithms and dividing by $2m$ gives $G_{2m}\leq G_ m+D/m$ with $D=\frac12\log \max (1,C_*J\mathrm e^{2C_ B})$. $\square$