Proposition 12.19.1894417AD14985

Proof. Although $t$ is defined only on the covering space $\Omega $, both $\mathrm{d}t$ and $\mathrm d^{\mathrm c}t$ descend to $X$. The Fubini–Study potentials $\log \| z\| ^2$ and $\log \| w\| ^2$ give $\mathrm{dd}^{\mathrm{c}}t=\beta -\alpha $. The forms $\alpha ,\beta $ are closed, and $\alpha ^2=\beta ^2=0$ since each is pulled back from a curve. As $D^2=0$, we obtain

In the last identity, the term involving $D$ contains $\mathrm{d}t$ twice, and $(\alpha +\beta )\wedge (\beta -\alpha )=0$. Thus $\omega $ is balanced, hence Gauduchon, and the formula for $\mathrm{dd}^{\mathrm{c}}\omega $ follows from that for $\mathrm{dd}^{\mathrm{c}}D$.

On a fiber, write $\zeta =a+\mathrm{i}b$, where both real periods are $2\pi $. There $t=2a$ plus a constant, so

Since $\omega ^3=6\alpha \wedge \beta \wedge D$, fiber integration and the normalization of the Fubini–Study forms give $\int _ X\omega ^3=24\pi $.

The identity (12.16.4) for Gauduchon metrics on threefolds, with $\mathrm{dd}^{\mathrm{c}}\omega =2\alpha \wedge \beta $, gives (12.19.19). Its last integral is nonnegative. Thus every smooth strictly $\omega $-plurisubharmonic potential has mass at most $24\pi $, and the constant potential attains this value. Regularization of bounded potentials, as in the proof of theorem 12.15.4, gives the same supremum over bounded potentials, so $v_+(\omega )=24\pi $. The metric independence in proposition 13.14.15 then gives the bounded mass property of $X$. The assertion about the lower mass is theorem 12.19.1. $\square$