Lemma 5.43.20.D7CDD8CAD2AC Let $\pi :Y\to X$ be a modification between connected compact complex manifolds, equipped with Hermitian forms $\omega _ Y,\omega _ X$. Write $\operatorname {Cap}_ Y=\operatorname {Cap}_{\omega _ Y}$ and $\operatorname {Cap}_ X=\operatorname {Cap}_{\omega _ X}$ for the Hermitian capacities of section 5.29. For quasi-plurisubharmonic functions $v_ j,v$ on $X$,
\[ v_ j\longrightarrow v\text{ in capacity on }X \quad \Longleftrightarrow \quad v_ j\circ \pi \longrightarrow v\circ \pi \text{ in capacity on }Y. \]
Proof.
Choose a proper analytic subset $S\subset X$ such that $\pi $ is an isomorphism outside $D=\pi ^{-1}(S)$. Both $S$ and $D$ have capacity zero. For $\epsilon >0$, choose an open neighborhood $V$ of $D$ of capacity less than $\epsilon $. By properness, there is a neighborhood $U_0$ of $S$ with $\pi ^{-1}(U_0)\subset V$; shrink it to an open set $U$ with $\operatorname {Cap}_ X(U)<\epsilon $. The inverse of $\pi $ is biholomorphic near the compact set $X\setminus U$. A finite cover by nested coordinate charts and the comparison of local and global capacities in section 5.29 give a constant $C_\epsilon $ such that, for every Borel set $E\subset X$,
\[ \begin{split} \operatorname {Cap}_ Y(\pi ^{-1}E)& \leq \epsilon +C_\epsilon \operatorname {Cap}_ X(E),\\ \operatorname {Cap}_ X(E)& \leq \epsilon +C_\epsilon \operatorname {Cap}_ Y(\pi ^{-1}E). \end{split} \]
Apply these inequalities to $E=\{ |v_ j-v|>\delta \} $ for each fixed $\delta >0$, omitting the pluripolar pole sets, and let first $j\to \infty $ and then $\epsilon \downarrow 0$.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.