Proposition 10.11.17F12058085DC

Proof. Choose a Gauduchon metric $\gamma $. For $T=\theta +\mathrm{dd}^{\mathrm{c}}u$ in $\alpha [-\varepsilon \omega ]$, the current $T+\varepsilon \omega $ is positive, and since $\mathrm{dd}^{\mathrm{c}}\gamma ^{n-1}=0$, Stokes' theorem for currents gives

The right-hand side does not depend on $T$. Since $\gamma $ is comparable to any fixed reference metric, this trace bounds all coefficient measures of the positive current $T+\varepsilon \omega $. Subtracting the fixed smooth form $\varepsilon \omega $ gives the asserted bound for $T$. In dimension one the same argument applies with $\gamma ^{n-1}=1$ and the ordinary Stokes theorem. $\square$