PomeProposition 12.17.9.A60EEF389BCA The split metric $g$ is balanced if and only if $\omega \wedge \alpha =\omega \wedge \beta =0$. For such a primitive pair one has $Q\leq 0$, with strict inequality wherever the pair is nonzero. In particular, a balanced split metric never satisfies the hypothesis $Q\geq 0$, $Q\not\equiv 0$ of theorem 12.17.7.
Proposition 12.17.9A60EEF389BCA
Proof. One has $\mathrm d(g^2)=2s\pi ^*\omega \wedge \mathrm de$, and $\mathrm de$ has one component containing $\eta $ and one containing $\bar\eta $. These two vertical types are independent, so $\mathrm d(g^2)=0$ if and only if $\omega \wedge F=0$, which is equivalent to the two stated real identities. A primitive real $(1,1)$-form on a surface has eigenvalues $\lambda ,-\lambda $ in an $\omega $-unitary frame, so its square is $-\lambda ^2\omega ^2$. Applying this to $\alpha $ and to $\beta $ gives the sign of $Q$. In this case $\mathrm{dd}^{\mathrm{c}}g\geq 0$, so (12.17.3) bounds the masses of bounded potentials from above rather than from below, and gives no information on the lower mass. $\square$
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