Theorem 12.6.32.40DF235D735C This span contains no nonzero smooth $\mathrm{dd}^{\mathrm{c}}$ direction, and its intersection with $\mathscr P$ is the cone
\[ \mathscr C=\{ b>0,\ e>0,\ e+cM>0\} . \]
For $\theta =\theta _{b,c,e}\in \mathscr C$, let $H(c,e)=\int _ B\sigma /L$. Then
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\begin{equation} \mathcal V_\mu (\theta )=\frac{nb^ m}{H(c,e)}. \end{equation}
A minimizing Gauduchon metric is
\[ \gamma =L^{-1/m}(A+Lw),\qquad A=b\kappa +\mathrm{dd}^{\mathrm{c}}\phi >0,\qquad A^ m=\frac{b^ m}{H}\frac{\sigma }{L}, \]
where $\phi $ is a smooth real function on $B$. This metric is balanced. Backgrounds with $c\ne 0$ are nonclosed.
Proof.
Pairing a smooth positive representative $\theta +\mathrm{dd}^{\mathrm{c}}u$ with the closed semipositive form $\kappa ^{m-1}\wedge w$ gives $b>0$, and its integral over the compact fiber $\{ y\} \times C$ is $L(y)>0$, by Stokes' theorem. These are exactly the stated inequalities; conversely, they make $\theta $ itself positive. The same tests applied to a smooth $\mathrm{dd}^{\mathrm{c}}$ direction give $b=0$ and $ca+e=0$ everywhere, and since $a$ is nonconstant, $c=e=0$. Also $\mathrm{d}\theta =c\, \mathrm{d}a\wedge w$, which is not identically zero when $c\ne 0$.
Next we apply theorem 14.2.1 on $B$ to the closed positive background $b\kappa $ and the density $(b^ m/H)\sigma /L$. Since this density has integral $b^ m=\int _ B(b\kappa )^ m$, Stokes' theorem forces the constant in the equation to be one, and we obtain $A$. Put $\Omega =A+Lw$. Since $A^{m+1}=0$ and $(Lw)^2=0$, the binomial expansion gives
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\begin{equation} \Omega ^ n=\frac{nb^ m}{H}\mu ,\qquad \gamma ^ m=L^{-1}A^ m+mA^{m-1}\wedge w. \end{equation}
The first summand is the pullback of a top-degree form on $B$, and the second is closed. Hence $\gamma $ is balanced, and in particular Gauduchon. For every Gauduchon metric $\eta $ on $X$, Stokes' theorem and the pointwise arithmetic–geometric mean inequality give
\[ \int _ X\theta \wedge \eta ^ m =\int _ X\Omega \wedge \eta ^ m \geq \left(\frac{nb^ m}{H}\right)^{1/n}D_\mu (\eta ). \]
For $\eta =\gamma $ the numerator is $nb^ m$, while $D_\mu (\gamma )=(nb^ m/H)^{m/n}H$. The quotient of $\gamma $ therefore attains the lower bound, which proves (12.6.33) and the minimizing property of $\gamma $.
$\square$
Pome
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