Corollary 18.26.22065C30217B81

Proof. By the current inequality $\omega +\mathrm{dd}^{\mathrm{c}}u\geq (1-Ct)\omega $, the potential $u+\varepsilon f$ is $\omega $-plurisubharmonic for both signs of $\varepsilon $, provided $|\varepsilon |$ is small: choose $c_ f>0$ with $-c_ f\omega \leq \mathrm{dd}^{\mathrm{c}}f\leq c_ f\omega $ and take $|\varepsilon |<(1-Ct)/c_ f$. These potentials differ boundedly from $u$, so proposition 18.26.21 gives the stated difference.

If $\mathrm{dd}^{\mathrm{c}}\omega =0$, the same proposition gives mass $A-Mt^2$ for every potential ordinarily equivalent to $u$. Conversely, write $\mathrm{dd}^{\mathrm{c}}\omega =\rho \omega ^2$ with $\rho $ smooth and real. If this form is not zero, choose $f=\rho $. Then $\int _ Xf\mathrm{dd}^{\mathrm{c}}\omega =\int _ X\rho ^2\omega ^2>0$, so the masses of the smooth perturbations above vary. Thus constancy of the mass forces $\mathrm{dd}^{\mathrm{c}}\omega =0$, which is the Gauduchon condition on a surface. $\square$