Theorem 12.17.7.7EC7E0C62DF9 Suppose $Q\geq 0$ and $Q\not\equiv 0$. The split form $g$ is Hermitian, and its lower mass is
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12.17.8
\begin{equation} v_-(g)=V=3sA_ E\int _ B\omega ^2. \end{equation}
Every bounded $u\in \mathrm{PSH}(X,g)$ satisfies
\[ \int _ X(g+\mathrm{dd}^{\mathrm{c}}u)^3 =V+\frac{3s}{2\pi }\int _ XE_ u\wedge \pi ^*Q\geq V. \]
Every Hermitian form $h\geq c g$, with $c>0$, satisfies (12.17.5). Thus the endpoint estimate of theorem 12.9.2 holds for every Hermitian form on $X$, with a constant depending on the form. The manifold $X$ admits no pluriclosed Hermitian metric.
Proof.
We suppress pullbacks in the curvature calculation. Comparing types in $\mathrm d\eta =F$ gives $\partial \eta =0$, $\bar\partial \eta =F$, $\partial \bar\eta =\bar F$ and $\bar\partial \bar\eta =0$. Hence
\[ \partial e=-\mathrm i\eta \wedge \bar F,\qquad \bar\partial e=\mathrm iF\wedge \bar\eta , \qquad \mathrm{dd}^{\mathrm{c}}e=-\frac1{2\pi }F\wedge \bar F=-\frac1{2\pi }Q. \]
Since $e^2=0$ and the base has dimension two,
\[ \mathrm{dd}^{\mathrm{c}}g=-\frac{s}{2\pi }\pi ^*Q,\qquad \mathrm{dd}^{\mathrm{c}}(g^2)=-\frac{s}{\pi }\pi ^*(\omega \wedge Q)=0, \qquad g^3=3s\pi ^*(\omega ^2)\wedge e. \]
The form $g$ is positive, because $\pi ^*\omega $ is positive on directions transverse to the fibers and $e$ is positive on the vertical line. Fiber integration of $g^3$ gives $V$, and lemma 12.17.2, lemma 12.17.4 give the mass identity and the endpoint bound. By compactness, every Hermitian form $h$ satisfies $h\geq cg$ for some $c>0$.
Finally, $S=-\mathrm{dd}^{\mathrm{c}}g$ is smooth, semipositive and not identically zero. If a Hermitian form $b$ were pluriclosed, Stokes' theorem would give
\[ 0=\int _ Xg\wedge \mathrm{dd}^{\mathrm{c}}b =\int _ Xb\wedge \mathrm{dd}^{\mathrm{c}}g=-\int _ Xb\wedge S<0, \]
where the last inequality holds because $b$ is strictly positive and $S$ is semipositive and nonzero on an open set.
$\square$
Pome
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