9141861B4F50
By proposition 12.3.2, the derivative of $\lambda _\mu $ at $\theta \in \mathscr P$ is represented by a minimizing Gauduchon metric, which is determined up to a constant factor. The solution form is conformal to this metric and has a prescribed volume form, so the value and the derivative of $\lambda _\mu $ together reconstruct the solution form, and hence the background class. Fix $\theta \in \mathscr P$ and a minimizing metric $\gamma _\theta $.
For $n\geq 2$ define the smooth positive $(n-1,n-1)$-form
\[ \Xi _\theta =\frac{\gamma _\theta ^{n-1}}{D_\mu (\gamma _\theta )}. \]
Since numerator and denominator are both homogeneous of degree $n-1$ in $\gamma _\theta $, this form does not depend on the choice of the minimizing metric. It is $\mathrm{dd}^{\mathrm{c}}$-closed, and by proposition 12.3.2 it represents the derivative in every smooth real $(1,1)$-direction $\chi $:
\[ (\mathrm{d}\lambda _\mu )_\theta (\chi )=\int _ X\chi \wedge \Xi _\theta . \]
Proposition 12.5.1.4BBB232A044C Let $n\geq 2$ and $\theta \in \mathscr P$. The value $\lambda _\mu (\theta )$ and the derivative of $\lambda _\mu $ at $\theta $ on all smooth real $(1,1)$-forms together determine the background class of $\theta $, and the derivative alone determines this class up to multiplication by a positive constant. More precisely, if $\widehat\gamma $ is the positive $(1,1)$-form with $\widehat\gamma ^{n-1}=\Xi _\theta $, then
\[ D_\mu (\widehat\gamma )=1, \qquad \Omega _\theta = \lambda _\mu (\theta ) \left(\frac{\mu }{\widehat\gamma ^ n}\right)^{1/n} \widehat\gamma . \]
Moreover, assigning to a background class in $\mathscr P$ with $\lambda _\mu =1$ the ray of its minimizing Gauduchon metrics is a bijection onto the set of rays of Gauduchon metrics.
Proof.
The values of the derivative on all smooth $(1,1)$-forms determine the smooth form $\Xi _\theta $. Its positive root exists and is unique by the positive-root construction preceding the proof of theorem 11.6.1. If $D=D_\mu (\gamma _\theta )$, this root is $\widehat\gamma =D^{-1/(n-1)}\gamma _\theta $, and the homogeneity of $D_\mu $ gives $D_\mu (\widehat\gamma )=1$.
By theorem 12.2.3, the solution form $\Omega _\theta $ is conformal to $\widehat\gamma $, and its volume form is $\lambda _\mu (\theta )^ n\mu $. These two requirements determine the conformal factor pointwise and give the formula for $\Omega _\theta $. Thus the value and the derivative recover $\Omega _\theta $, a representative of the class of $\theta $. Without the value, the same formula recovers $\Omega _\theta $ up to a positive constant factor. Conversely, multiplying the class by a positive constant multiplies $\Omega _\theta $ by the same constant and leaves its minimizing Gauduchon ray, and hence $\Xi _\theta $, unchanged; so the derivative cannot determine more than the ray of the class.
For the surjectivity in the last assertion, take any Gauduchon metric $\gamma $ and set
\[ \Omega _\gamma =\left(\frac\mu {\gamma ^ n}\right)^{1/n}\gamma . \]
This Hermitian form is unchanged by constant rescaling of $\gamma $ and satisfies $\Omega _\gamma ^ n=\mu $. The zero potential therefore solves the equation for the background $\Omega _\gamma $ with constant one, so its class has $\lambda _\mu =1$, and $\gamma $ is a minimizing Gauduchon metric by theorem 12.2.3. Injectivity follows from the recovery formula: two classes with $\lambda _\mu =1$ and the same Gauduchon ray have the same solution form. This proves the bijection.
$\square$
In dimension one, a background class is determined by its integral, by the Green operator of the curve. The derivative is then the fixed functional $\chi \mapsto \int _ X\chi $, and the value alone determines the class.
Pome
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