Theorem 12.16.3.26629526444F Assume $n=3$ and $v\in \mathcal H_\omega $. The following are equivalent:
$v$ is a local minimum of $\mathrm M_\omega $;
$v$ is a global minimum of $\mathrm M_\omega $ on $\mathcal H_\omega $;
$\Omega =\omega +\mathrm{dd}^{\mathrm{c}}v$ is Gauduchon and $\mathrm{dd}^{\mathrm{c}}\omega \leq 0$.
In this case $v_-(\omega )=\int _ X\Omega ^3$, and the infimum in (12.15.5) is attained. The corresponding statements for maxima hold with $\mathrm{dd}^{\mathrm{c}}\omega \geq 0$, and then $v_+(\omega )=\int _ X\Omega ^3<+\infty $.
Proof.
Since $\mathrm{dd}^{\mathrm{c}}\Omega =\mathrm{dd}^{\mathrm{c}}\omega $, the preceding proposition shows that the first condition implies the third. Conversely, if $\Omega $ is Gauduchon, expanding the cube and integrating by parts gives, for every smooth real $f$,
D5CA82478984
12.16.4
\begin{equation} \int _ X(\Omega +\mathrm{dd}^{\mathrm{c}}f)^3-\int _ X\Omega ^3 =-3\int _ X\mathrm{d}f\wedge \mathrm d^{\mathrm c}f\wedge \mathrm{dd}^{\mathrm{c}}\omega . \end{equation}
The form $\mathrm{d}f\wedge \mathrm d^{\mathrm c}f$ is nonnegative, so the right side is nonnegative if $\mathrm{dd}^{\mathrm{c}}\omega \leq 0$; then $v$ is a global minimum, and in particular a local one. By theorem 12.15.4, regularization identifies the infimum of the masses of smooth strictly $\omega $-plurisubharmonic potentials with $v_-(\omega )$, and proposition 12.15.6 proves attainment in (12.15.5). Reversing the sign in the same identity proves the statements for maxima, including the equality of $\int _ X\Omega ^3$ with the supremum over bounded potentials.
$\square$
Pome
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