Proposition 12.4.1.67FDDBC10DBD Along this affine variation,
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12.4.2
\begin{align} \frac{\ddot\lambda }{\lambda } & =-\frac1n\int _ X \left\| A_\chi -\frac{\dot\lambda }{\lambda }I\right\| _\Omega ^2 \, \mathrm{d}\nu _\Omega ,\\ -\frac{\mathrm{d}^2}{\mathrm{d}t^2}\log \lambda _\mu (\theta _ t) & =\frac1n\int _ X\operatorname {tr}(A_\chi ^2)\, \mathrm{d}\nu _\Omega . \tag{12.4.3} \end{align}
The first expression vanishes exactly when $\chi =r\theta +\mathrm{dd}^{\mathrm{c}}f$ for a real constant $r$ and a smooth real $f$. The second vanishes exactly when $\chi =\mathrm{dd}^{\mathrm{c}}f$ for a smooth real $f$.
Proof.
Smooth dependence on $t$ was proved in proposition 12.3.2. We differentiate the logarithmic form $\log (\Omega _ t^ n/\mu )=n\log \lambda _\mu (\theta _ t)$ of the equation. Since $\mu $ is fixed,
\[ \operatorname {tr}A_\chi =n\frac{\dot\lambda }{\lambda }. \]
In particular this trace is constant on $X$. Differentiating once more along the affine path $\theta _ t$ gives
\[ \operatorname {tr}_\Omega (\mathrm{dd}^{\mathrm{c}}\ddot u) -\operatorname {tr}(A_\chi ^2) =n\left(\frac{\ddot\lambda }{\lambda } -\frac{\dot\lambda ^2}{\lambda ^2}\right). \]
Integration against the probability $\nu _\Omega $, which is fixed since everything is evaluated at $t=0$, removes the first term. This proves (12.4.3).
Put $r=\dot\lambda /\lambda $. Since $A_\chi $ is Hermitian and $\operatorname {tr}A_\chi =nr$,
\[ \| A_\chi -rI\| _\Omega ^2 =\operatorname {tr}(A_\chi ^2)-nr^2. \]
Substituting this into (12.4.3) gives (12.4.2). Since the adjoint measure has a strictly positive smooth density, the right side of (12.4.2) vanishes exactly when $A_\chi =rI$ everywhere, that is, when
\[ \chi +\mathrm{dd}^{\mathrm{c}}\dot u=r(\theta +\mathrm{dd}^{\mathrm{c}}u). \]
This is the stated relation with $f=ru-\dot u$. Conversely, if $\chi =r\theta +\mathrm{dd}^{\mathrm{c}}f$, then $\theta _ t$ differs from $(1+rt)\theta $ by a smooth $\mathrm{dd}^{\mathrm{c}}$ term for small $t$. Homogeneity gives $\lambda _\mu (\theta _ t)=(1+rt)\lambda $, whose second derivative is zero.
The integrand in (12.4.3) is the squared norm of the Hermitian endomorphism $A_\chi $. Its integral therefore vanishes exactly when $A_\chi =0$, that is, when $\chi =-\mathrm{dd}^{\mathrm{c}}\dot u$. Conversely, $\lambda _\mu (\theta +t\mathrm{dd}^{\mathrm{c}}f)=\lambda $ for all $t$. This proves both equality statements.
$\square$
Pome
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