Proposition 12.21.1.DE2BA13B1A92 For $\epsilon >0$, put $g=q+\epsilon $ and $I=\int _ Eg^{-1}\omega _1$. Then
5C3988BE0A59
12.21.2
\begin{equation} \mathcal V_\mu (\theta +\epsilon h) =\frac{2}{I}\left(\int _ E(a+\epsilon )\omega _1- \int _ E\frac{|r|^2}{g}\omega _1\right), \end{equation}
and the infimum defining this volume is attained by a smooth positive Gauduchon metric. In particular, if $a$ is a constant $b>0$ and $r=0$, then $\mathcal V_\mu (\theta +\epsilon h)=2(b+\epsilon )/I$. If $|r|^2=aq$, then
A72D1E62A49B
12.21.3
\begin{equation} \mathcal V_\mu (\theta +\epsilon h)=\frac{2\epsilon (1+K)}{I}, \qquad K=\int _ E\frac{a}{q+\epsilon }\omega _1. \end{equation}
Proof.
Define
\[ c=I^{-1}\left(\int _ E(a+\epsilon )\omega _1- \int _ E\frac{|r|^2}{g}\omega _1\right),\qquad A=\frac{c+|r|^2}{g}. \]
Since $(a+\epsilon )g-|r|^2>0$, we have $c>0$, and the definition of $c$ makes $A-a-\epsilon $ have zero mean with respect to $\omega _1$. Dividing its nonconstant Fourier coefficients by the corresponding nonzero eigenvalues of the flat Laplacian produces a smooth real function $u$ on $E$ with $\mathrm{dd}^{\mathrm{c}}u=(A-a-\epsilon )\omega _1$. Pull it back to $X$. The form $\Omega =\theta +\epsilon h+\mathrm{dd}^{\mathrm{c}}u$ has coefficient matrix $\left(\begin{smallmatrix}A& r\\ \bar r& g\end{smallmatrix}\right)$, which has determinant $c$ and positive diagonal entries. Hence $\Omega >0$ and $\Omega ^2=2c\mu $, and theorem 12.2.3 gives (12.21.2).
The metric $\gamma _*=\Omega /g$ has constant coefficient $1$ in the fiber direction, and its remaining coefficients depend only on $z$. Applying $\mathrm{dd}^{\mathrm{c}}$ to its base term or its mixed terms produces a repeated $\mathrm{d} z$ or $\mathrm{d}\bar z$, so $\mathrm{dd}^{\mathrm{c}}\gamma _*=0$. Thus $\gamma _*$ is a smooth Gauduchon metric conformal to $\Omega $, and by theorem 12.2.3 it attains the infimum. The two special cases follow by direct substitution.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.