Proposition 11.2.17.AE6CE389C569 Let $\omega $ be Hermitian symplectic. For every smooth real function $\varphi $ with $\omega _\varphi =\omega +\mathrm{dd}^{\mathrm{c}}\varphi >0$, one has $\rho _{\omega _\varphi }=\rho _\omega $ and
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11.2.18
\begin{equation} F(\omega _\varphi )+V(\omega _\varphi )=F(\omega )+V(\omega ). \end{equation}
Moreover, $DF_\omega (\mathrm{dd}^{\mathrm{c}}\varphi )=0$ for every smooth real $\varphi $ if and only if $\mathrm{dd}^{\mathrm{c}}(\omega ^3)=0$. Thus a critical point of $F$ in its Aeppli class is Gauduchon.
Proof.
Fix any completion $\zeta $ of $\omega $ and put $\Theta _\zeta =\omega +\zeta +\bar\zeta $. Expansion by type gives
\[ \frac1{4!}\int _ X\Theta _\zeta ^4 =V(\omega )+\frac12\int _ X\zeta \wedge \bar\zeta \wedge \omega ^2 +\frac14\int _ X\zeta ^2\wedge \bar\zeta ^2. \]
Since $\partial \omega _\varphi =\partial \omega $, the forms $\omega _\varphi $ and $\omega $ have the same completions. With $\zeta $ fixed, replacing $\omega $ by $\omega _\varphi $ changes the closed form $\Theta _\zeta $ by an exact form, so the integral on the left is unchanged; so is the last integral on the right, which does not involve $\omega $. Hence
\[ \| \zeta \| _{L^2(\omega _\varphi )}^2 -\| \zeta \| _{L^2(\omega )}^2=V(\omega )-V(\omega _\varphi ). \]
This difference is independent of $\zeta $, so the two norms have the same minimizer, which gives $\rho _{\omega _\varphi }=\rho _\omega $ and (11.2.18). Differentiating at $t=0$ along $\omega +t\mathrm{dd}^{\mathrm{c}}\varphi $ and using lemma 11.1.2, we obtain
\[ DF_\omega (\mathrm{dd}^{\mathrm{c}}\varphi )=-\frac16\int _ X\varphi \mathrm{dd}^{\mathrm{c}}(\omega ^3). \]
Since $\varphi $ is arbitrary, this proves the remaining assertions.
$\square$
Pome
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