PomeLemma 19.9.12.976C06708A77 If a compact complex manifold $X$ admits a modification $\pi :Z\to X$ from a compact Kähler manifold, then $X$ has the bounded mass property.
Lemma 19.9.12976C06708A77
Proof. Fix a Hermitian form $\omega $ on $X$ and a Kähler form $\kappa $ on $Z$, and choose $C>0$ with $\pi ^*\omega \leq C\kappa $. By proposition 13.14.15, it suffices to bound the masses of smooth functions $u\in \mathrm{PSH}(X,\omega )$. Their pullbacks satisfy
\[ 0\leq \pi ^*\omega +\mathrm{dd}^{\mathrm{c}}(u\circ \pi ) \leq C\kappa +\mathrm{dd}^{\mathrm{c}}(u\circ \pi ). \]
Taking top powers and integrating gives
\[ \begin{split} \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^ n & =\int _ Z(\pi ^*\omega +\mathrm{dd}^{\mathrm{c}}(u\circ \pi ))^ n\\ & \leq \int _ Z(C\kappa +\mathrm{dd}^{\mathrm{c}}(u\circ \pi ))^ n =C^ n\int _ Z\kappa ^ n. \end{split} \]
The first equality follows by change of variables on the set where $\pi $ is biholomorphic, since smooth top-degree forms give the exceptional analytic sets zero mass. The inequality follows by expanding in the smooth semipositive difference $C\kappa -\pi ^*\omega $, and the last equality follows from Stokes' theorem because $\kappa $ is closed. The bound does not depend on $u$, which proves the bounded mass property. $\square$
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