Proposition 11.1.8.5C29873687ED For smooth real functions $u,v$ on $X$,
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11.1.9
\begin{equation} \left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\mathrm M_\omega (u+tv) =n\int _ X v\mathrm{dd}^{\mathrm{c}}(\omega _ u^{n-1}). \end{equation}
If $\mathrm M_\omega $ is constant on $\mathcal H_\omega $, then $\mathrm{dd}^{\mathrm{c}}(\omega _ u^{n-1})=0$ for every $u\in \mathcal H_\omega $.
Proof.
Since $\mathrm M_\omega (u+tv)$ is a polynomial in $t$, its derivative at $t=0$ is
\[ n\int _ X\mathrm{dd}^{\mathrm{c}}v\wedge \omega _ u^{n-1}, \]
and lemma 11.1.2 turns this into the right-hand side of (11.1.9). For the second assertion, fix $u\in \mathcal H_\omega $. For every smooth real direction $v$, the segment $u+tv$ lies in $\mathcal H_\omega $ when $|t|$ is sufficiently small, so the right-hand side of (11.1.9) vanishes for all smooth real $v$. Hence the smooth top-degree form $\mathrm{dd}^{\mathrm{c}}(\omega _ u^{n-1})$ vanishes.
$\square$
Pome
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