Proposition 20.4.22.D7373D3639FA Let $g:Z\to Y$ be a smooth compact modification, and fix a Gauduchon metric $\gamma '$ on $Z$. Pullback of positive closed $(1,1)$-currents, defined through their local potentials, is weakly continuous. Moreover, there is a constant $C_ g$, depending only on the modification and the fixed metrics, such that
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20.4.23
\begin{equation} \int _ Zg^*T\wedge (\gamma ')^{n-1} \leq C_ g\int _ YT\wedge \gamma ^{n-1} \qquad (T\geq 0,\ \mathrm{d}T=0). \end{equation}
Proof.
Write $T=B_\gamma (T)+\mathrm{dd}^{\mathrm{c}}v_\gamma (T)$. The pullback is
\[ g^*T=g^*B_\gamma (T)+\mathrm{dd}^{\mathrm{c}}(v_\gamma (T)\circ g). \]
This current is positive and closed, since it is locally $\mathrm{dd}^{\mathrm{c}}$ of plurisubharmonic functions, and the composed potential is not identically $-\infty $ because $g$ is surjective. The formula agrees with the definition through local potentials, so it does not depend on the decomposition.
If $T_ j\to T$, the smooth parts converge in $C^\infty $, and the potentials converge in $L^1(Y)$ with a common lower curvature bound. By lemma 5.43.19, their compositions with $g$ converge in $L^1(Z)$, and taking $\mathrm{dd}^{\mathrm{c}}$ proves the weak convergence of the pullbacks.
Integration by parts on $Z$ gives
\[ \int _ Zg^*T\wedge (\gamma ')^{n-1} =\int _ Zg^*B_\gamma (T)\wedge (\gamma ')^{n-1}. \]
The absolute value of the right-hand side is at most a fixed constant times $\| B_\gamma (T)\| _{C^0}$. Since the left-hand side is nonnegative, (20.4.21) gives (20.4.23).
$\square$
Pome
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