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Differentiating the smooth equation twice along an affine family of backgrounds computes the second derivative of the root volume. After integration against the adjoint probability of the solution form, the second derivative of the potential disappears, and what remains is the integrated squared norm of the linearized solution form. Consequently the Hessian of $-\log \lambda _\mu $ is positive definite once smooth $\mathrm{dd}^{\mathrm{c}}$ directions are factored out. On the cone of closed Kähler classes, where $\lambda _\mu (\theta )=(\int _ X\theta ^ n)^{1/n}$, it is $1/n$ times the Hessian of $-\log \int _ X\theta ^ n$.
Fix $\theta \in \mathscr P$ and write $\Omega =\Omega _\theta $, $u=u_\theta $ and $\lambda =\lambda _\mu (\theta )$. Let $\nu _\Omega $ be the adjoint probability of $\Omega $ from proposition 11.23.2, that is, the smooth positive probability measure, given by the Gauduchon multiplier of $\Omega $, which satisfies $\int _ X\operatorname {tr}_\Omega (\mathrm{dd}^{\mathrm{c}}v)\, \mathrm{d}\nu _\Omega =0$ for every smooth real $v$. In dimension one this probability is $\Omega /\int _ X\Omega =\mu $.
Let $\theta _ t=\theta +t\chi $, where $t$ is real and $\chi $ is a smooth real $(1,1)$-form on $X$. For sufficiently small $|t|$, the background $\theta _ t$ stays in $\mathscr P$. We normalize the solution potentials $u_ t$ by one fixed condition on their mean, and dots denote derivatives of $u_ t$ and of $\lambda _\mu (\theta _ t)$ at $t=0$. Define the Hermitian endomorphism
\[ A_\chi =\Omega ^{-1}(\chi +\mathrm{dd}^{\mathrm{c}}\dot u) \]
of $T^{1,0}X$, where $\Omega ^{-1}$ means raising an index with the positive form $\Omega $; traces of powers of $A_\chi $ are independent of coordinates. We write $I$ for the identity endomorphism of $T^{1,0}X$ and $\| \cdot \| _\Omega $ for the Hilbert–Schmidt norm on its endomorphisms induced by $\Omega $.
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.
$\square$
The linearized solution and the derivative of the constant depend linearly on $\chi $. For smooth real $(1,1)$-forms $\chi _1,\chi _2$ on $X$, polarization therefore gives a symmetric bilinear form
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12.4.4
\begin{equation} \mathfrak g_\theta (\chi _1,\chi _2) =-D^2\log \lambda _\mu |_\theta (\chi _1,\chi _2) =\frac1n\int _ X\operatorname {tr}(A_{\chi _1}A_{\chi _2})\, \mathrm{d}\nu _\Omega . \end{equation}
We call the classes of smooth real $(1,1)$-forms modulo $\mathrm{dd}^{\mathrm{c}}C^\infty (X,\mathbb R)$ background classes. By proposition 12.4.1, the form $\mathfrak g_\theta $ descends to background classes of directions and is positive definite there. A change of representative of $\theta $ leaves its solution form $\Omega $ unchanged, and a change of representative of a direction is absorbed by the derivative of the solution potential. Thus $\mathfrak g$ is intrinsically defined on the background classes in $\mathscr P$. On every finite-dimensional affine family whose directions are linearly independent modulo smooth $\mathrm{dd}^{\mathrm{c}}$ terms, it is a smooth Riemannian metric.
These variations change the background class. Directions obtained by varying a potential inside one fixed class are exactly the directions annihilated by (12.4.4). Thus this construction concerns a different space from the potential-space geometry of section 11.2.
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