Proposition 12.6.547DDBF953B9C

Proof. Invariant averages and balancedness. Let $Z_1,\ldots ,Z_ n$ be the invariant holomorphic vector fields dual to $\alpha _1,\ldots ,\alpha _ n$, and set $\Upsilon =\alpha _1\wedge \cdots \wedge \alpha _ n$. This holomorphic top-degree form and its conjugate are closed. For an invariant holomorphic vector field $Z$, evaluation on the invariant frame gives

The second identity also follows from Cartan's formula. Since $h_0^ n$ is a positive constant multiple of the appropriately oriented form $\Upsilon \wedge \bar\Upsilon $, Stokes' theorem on compact $X$ gives

Thus the fields $Z_ j$ and their conjugates have zero divergence with respect to $\mu $. In particular,

Cartan's formula also gives $\mathrm d(\iota _{Z_ j}\Upsilon )=0$. These contractions span the invariant $(n-1,0)$-forms. The form $\theta _ A^{n-1}$ is a constant linear combination of their wedges with conjugates, so $\mathrm d\theta _ A^{n-1}=0$.

The holomorphic fields commute with the antiholomorphic fields, and $\mathrm d\alpha _ j$ has type $(2,0)$. The coefficient matrix of $\mathrm{dd}^{\mathrm{c}}u$ is therefore $(Z_ j\bar Z_ k u)$. For smooth $u$ its entries have $\mu $-mean zero by (12.6.8), so the matrix of $\mu $-means of the coefficients of $\theta _ A+\mathrm{dd}^{\mathrm{c}}u$ is $A$. If $\theta _ A+\mathrm{dd}^{\mathrm{c}}u>0$, averaging the positive coefficient matrix gives $A>0$. The converse follows by taking $u=0$. The same average shows that $V_ W$ contains no nonzero smooth $\mathrm{dd}^{\mathrm{c}}$ direction.

For a distribution $u$, the adjoints of $Z_ j$ and $\bar Z_ k$ with respect to $\mu $ annihilate constants, so the mean of each $Z_ j\bar Z_ k u$ is still zero. The coefficients of a positive current in this global frame form a positive matrix of measures, whose matrix of total masses is semipositive; for $\theta _ A+\mathrm{dd}^{\mathrm{c}}u\geq 0$ this gives $A\geq 0$. Conversely, $\theta _ A$ itself is a positive current when $A\geq 0$. If $A=0$, the positive trace measure of $\mathrm{dd}^{\mathrm{c}}u$ has total mass zero, and polarization then makes the entire current zero. Finally, for a fixed positive $A_+\in W$, the matrices $A+\varepsilon A_+$ with $\varepsilon \downarrow 0$ approximate every semipositive $A\in W$ from the positive cone. This proves the closure assertion.

The quotient and its equality case. Put $Q=H\det A$, so that $\theta _ A^ n=Q\mu $. For every smooth positive Hermitian form $\gamma $, the matrix arithmetic–geometric mean inequality gives

Integration yields $\int _ X\theta _ A\wedge \gamma ^{n-1}\geq Q^{1/n}D_\mu (\gamma )$, which bounds every Gauduchon quotient in definition 12.2.1 from below. The balanced, hence Gauduchon, metric $\gamma =\theta _ A$ has $D_\mu (\theta _ A)=Q^{(n-1)/n}$ and attains equality. This proves the formula for $\lambda _\mu (\theta _ A)$.

For a minimizing Gauduchon metric $\gamma $, the continuous nonnegative difference of the two sides of the pointwise inequality has integral zero. Pointwise equality then forces $\gamma =f\theta _ A$ for a smooth positive function $f$. Set $q=f^{n-1}$ and $T=\theta _ A^{n-1}$. Since $\mathrm dT=0$, the Gauduchon equation becomes $\mathrm{dd}^{\mathrm{c}}q\wedge T=0$, and

The last integrand is nonnegative and vanishes exactly where $\mathrm dq=0$. Thus $f$ is constant. As $n\geq 2$ and

normalization determines $c=\lambda _\mu (\theta _ A)^{-1}$. This proves the unique minimizer and its volume identity.

Differentiating $-\log \lambda _\mu (\theta _ A)=-(\log H+\log \det A)/n$ twice gives (12.6.7). The corresponding squared norm of $U$ is $\| A^{-1/2}UA^{-1/2}\| _{\mathrm F}^2/n$, which is positive for $U\ne 0$; here $\| B\| _{\mathrm F}^2=\operatorname {tr}(B^*B)$ denotes the squared Frobenius norm.

Finally,

For invertible $A$, the vanishing of this form forces every $\mathrm d\alpha _ j=0$, by independence of the conjugate coframe, and the Lie algebra would then be abelian. Since adding $\mathrm{dd}^{\mathrm{c}}u$ does not change $\mathrm d\theta _ A$, this proves the last assertion. $\square$