PomeLemma 5.29.22.E8D07C38BC7C For a smooth real $(1,1)$-form $\sigma $ on $Y$, put $A=\int _ Y\sigma $. There is a smooth real function $h$ with $\sigma =A\eta +\mathrm{dd}^{\mathrm{c}}h$. The class $\mathrm{PSH}(Y,\sigma )$ is nonempty if and only if $A\geq 0$. If $A=0$, it consists of the functions $-h+c$, $c\in \mathbb R$, and its unique member with supremum zero is $u_\sigma =-h+\min _ Yh$.
Lemma 5.29.22E8D07C38BC7C
Proof. The smooth form $\sigma -A\eta $ has integral zero, so the Poisson equation on the compact connected surface $Y$ has a smooth solution $h$, unique up to an additive constant. If $u\in \mathrm{PSH}(Y,\sigma )$, positivity and Stokes' theorem give
\[ 0\leq (\sigma +\mathrm{dd}^{\mathrm{c}}u)(Y)=A. \]
Conversely, $\sigma +\mathrm{dd}^{\mathrm{c}}(-h)=A\eta $, so $-h\in \mathrm{PSH}(Y,\sigma )$ when $A\geq 0$. If $A=0$, every positive curvature measure has total mass zero and therefore vanishes. Thus $\mathrm{dd}^{\mathrm{c}}(u+h)=0$ for every $u\in \mathrm{PSH}(Y,\sigma )$, and elliptic regularity and the uniqueness of harmonic functions on $Y$ show that $u+h$ is constant. Normalizing the supremum gives the stated formula. $\square$
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