F6A1B32989A1
As for the Monge–Ampère equation, solving the linearized Gauduchon volume equation requires a compatibility condition, given by an adjoint probability measure. Taking the $(n-1)$st root multiplies the trace linearization by the factor $n/(n-1)$, and the first-order term of the ansatz introduces a drift term. We compute the linearized operator at any potential with $\Gamma _ u>0$, and then express its adjoint condition as an equation for a positive $(1,1)$-form. Throughout this section we put $K=\gamma ^{n-2}$.
Proposition 14.12.1.706EB82AB6A2 On the open set of smooth potentials $u$ with $\Gamma _ u>0$, the derivative of
\[ \mathcal G(u)=\log \frac{\eta _ u^ n}{\gamma ^ n} \]
is the real elliptic operator
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14.12.2
\begin{equation} \mathcal L_ uv= \frac{n}{n-1}\frac{\eta _ u\wedge \mathcal D_\gamma v}{\eta _ u^ n}. \end{equation}
It has no zeroth-order term, and its kernel consists of the constants. There is a unique smooth positive function $\rho _ u$, normalized by $\int _ X\rho _ u\eta _ u^ n=1$, such that the probability measure
\[ \mathrm{d}\mu _ u^\gamma =\rho _ u\eta _ u^ n \]
satisfies $\int _ X\mathcal L_ uv\, \mathrm{d}\mu _ u^\gamma =0$ for every smooth real $v$.
Proof.
For a smooth path of potentials $u(t)$, write $\eta =\eta _{u(t)}$ and $\Gamma =\Gamma _{u(t)}$. Differentiating $\eta ^{n-1}=\Gamma $ gives $(n-1)\dot\eta \wedge \eta ^{n-2}=\dot\Gamma $, where $\dot\Gamma =\mathcal D_\gamma \dot u$. Wedging with $\eta $ and dividing by $\eta ^ n$, we obtain
\[ \frac{\mathrm{d}}{\mathrm{d}t}\log \eta ^ n =\frac{n}{n-1}\frac{\eta \wedge \dot\Gamma }{\eta ^ n}, \]
which proves (14.12.2).
Only the second-order term $\mathrm{dd}^{\mathrm{c}}v\wedge \gamma ^{n-2}$ of $\mathcal D_\gamma v$ contributes to the principal symbol. For a nonzero complex covector $\zeta $ of type $(1,0)$, the form
\[ \eta _ u\wedge \mathrm i\zeta \wedge \overline\zeta \wedge \gamma ^{n-2} \]
is a positive volume form. Hence the principal symbol is positive definite, and by compactness the operator is uniformly elliptic for each fixed $u$. The remaining term is of first order, and constants are annihilated, so the strong maximum principle gives $\ker \mathcal L_ u=\mathbb R$.
The Fredholm and positivity arguments in the proof of theorem 11.4.2 apply to this real elliptic operator with first-order terms and no zeroth-order term. Its index is zero by deformation to a Laplacian, so its formal adjoint with respect to $\eta _ u^ n$ has a one-dimensional kernel spanned by a smooth strictly positive function. Normalizing its integral to one gives $\rho _ u$ and proves uniqueness.
$\square$
The adjoint condition can be expressed as an equation for the positive $(1,1)$-form $\rho _ u\eta _ u$. For a smooth real $(1,1)$-form $A$, set
\[ \mathcal A_\gamma (A) =\mathrm{dd}^{\mathrm{c}}A\wedge \gamma ^{n-2} +\operatorname {Re}\left(\frac{\mathrm i}{2\pi } \partial A\wedge \bar\partial (\gamma ^{n-2})\right). \]
This is a real form of top degree, and $\mathcal A_\gamma $ is the formal adjoint of $\mathcal D_\gamma $ under the pairing of $(1,1)$-forms with $(n-1,n-1)$-forms, as the proof below shows.
Proposition 14.12.3.029A1D30A55E The function $\rho _ u$ of proposition 14.12.1 is characterized by
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14.12.4
\begin{equation} \mathcal A_\gamma (\rho _ u\eta _ u)=0, \qquad \int _ X\rho _ u\eta _ u^ n=1. \end{equation}
When $n=2$, its first equation is $\mathrm{dd}^{\mathrm{c}}(\rho _ u\eta _ u)=0$. When $\eta _ u=\gamma $, the function $\rho _ u$ is constant and $\mathrm{d}\mu _ u^\gamma =\gamma ^ n/\int _ X\gamma ^ n$.
Proof.
We first prove that, for every smooth real $(1,1)$-form $A$ and every smooth real function $v$ on $X$,
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14.12.5
\begin{equation} \int _ X A\wedge \mathcal D_\gamma v =\int _ X v\, \mathcal A_\gamma (A). \end{equation}
Write $c=\mathrm i/(2\pi )$. Since $A,K,v$ are real, expanding the real part gives
\[ \mathcal D_\gamma v =c\, \partial \bar\partial v\wedge K +\frac c2\bigl (\partial v\wedge \bar\partial K -\bar\partial v\wedge \partial K\bigr ). \]
Integrating the first term by parts twice gives $\int _ Xv\, c\partial \bar\partial (A\wedge K)$, and integrating each first-order term by parts once gives the correction
\[ \int _ Xv\, \frac c2 \bigl (-\partial A\wedge \bar\partial K +\bar\partial A\wedge \partial K -2A\wedge \partial \bar\partial K\bigr ). \]
Finally,
\[ \partial \bar\partial (A\wedge K) =\partial \bar\partial A\wedge K -\bar\partial A\wedge \partial K +\partial A\wedge \bar\partial K +A\wedge \partial \bar\partial K. \]
Adding these contributions cancels the terms containing $A\wedge \partial \bar\partial K$ and leaves $\mathcal A_\gamma (A)$. This proves (14.12.5). Applied to $A=\rho _ u\eta _ u$, it shows by (14.12.2) that $\int _ X\mathcal L_ uv\, \mathrm{d}\mu _ u^\gamma $ is a fixed positive multiple of $\int _ Xv\, \mathcal A_\gamma (\rho _ u\eta _ u)$, so the adjoint condition is equivalent to (14.12.4).
For $n=2$, we have $K=1$, which gives the first special case. For the second, the product rule gives
\[ \mathcal A_\gamma (\gamma ) =\frac1{n-1}\mathrm{dd}^{\mathrm{c}}(\gamma ^{n-1})=0. \]
By uniqueness, $\rho _ u$ is consequently the constant $(\int _ X\gamma ^ n)^{-1}$.
$\square$
For $n>2$ the equation for $\rho _ u\eta _ u$ involves the first derivatives of $\gamma ^{n-2}$, so the fixed auxiliary metric $\gamma $ remains part of the adjoint equation; it is not simply the Gauduchon condition for $\eta _ u$.
Pome
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