Theorem 17.16.12.7A7AE56CA513 Let $u_ t\in \mathcal H_\omega $, $0\leq t\leq 1$, be a smooth path, and put $\eta _ t=\omega _{u_ t}$ and $T_ t=\eta _ t^{n-1}$. The path is stationary for $\mathscr S$ if and only if
DCC8DA6D179E
17.16.13
\begin{equation} \begin{split} \ddot u_ t\, \eta _ t^ n={}& n\, \mathrm{d}\dot u_ t\wedge \mathrm d^{\mathrm c}\dot u_ t\wedge T_ t\\ & +n\dot u_ t\bigl (\mathrm{d}\dot u_ t\wedge \mathrm d^{\mathrm c}T_ t -\mathrm d^{\mathrm c}\dot u_ t\wedge \mathrm{d}T_ t\bigr ) +\frac n2\dot u_ t^{\, 2}\mathrm{dd}^{\mathrm{c}}T_ t. \end{split} \end{equation}
All differentials on the right act only in the $X$ variable.
Proof.
Vary the path as $u_ t+s v_ t$, where $v_ t$ is smooth in $(x,t)$ and $v_0=v_1=0$, and put $f=\dot u_ t$. Differentiating the action at $s=0$, using $\partial _ s(\omega _{u_ t+sv_ t}^ n)|_{s=0}=n\mathrm{dd}^{\mathrm{c}}v_ t\wedge T_ t$, gives
\[ \delta \mathscr S =\int _0^1\left(\int _ X f\dot v_ t\, \eta _ t^ n +\frac n2\int _ X f^2\mathrm{dd}^{\mathrm{c}}v_ t\wedge T_ t\right)\mathrm{d}t. \]
We integrate by parts in $t$ in the first term, using $\partial _ t(\eta _ t^ n)=n\mathrm{dd}^{\mathrm{c}}f\wedge T_ t$, and on $X$ in the second, using lemma 11.1.2 with the whole form $f^2T_ t$. Thus
\[ \delta \mathscr S =\int _0^1\int _ Xv_ t\left(-\ddot u_ t\eta _ t^ n -nf\mathrm{dd}^{\mathrm{c}}f\wedge T_ t+\frac n2\mathrm{dd}^{\mathrm{c}}(f^2T_ t)\right)\mathrm{d}t. \]
The product rule gives
\[ \begin{split} \mathrm{dd}^{\mathrm{c}}(f^2T_ t)={}& 2f\mathrm{dd}^{\mathrm{c}}f\wedge T_ t +2\mathrm{d}f\wedge \mathrm d^{\mathrm c}f\wedge T_ t\\ & +2f\bigl (\mathrm{d}f\wedge \mathrm d^{\mathrm c}T_ t-\mathrm d^{\mathrm c}f\wedge \mathrm{d}T_ t\bigr ) +f^2\mathrm{dd}^{\mathrm{c}}T_ t. \end{split} \]
After the term $nf\mathrm{dd}^{\mathrm{c}}f\wedge T_ t$ cancels, the integrand is $v_ t$ times the difference between the right-hand and left-hand sides of (17.16.13). Since $\eta _ t>0$ on the compact set $X\times [0,1]$, every smooth $v_ t$ gives an admissible variation for small $|s|$, and the fundamental lemma of the calculus of variations gives both implications.
$\square$
Pome
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