Proposition 9.11.65D0F33FF75830

Proof. Choose real smooth functions $A$ on $C$, supported away from $q$, and $B$ on $E_2$. Let

The form $\alpha $ is real and satisfies $\mathrm{dd}^{\mathrm{c}}\alpha =0$. At a point where $\partial A=0$, $\partial \bar\partial A\ne 0$ and $\partial B\ne 0$,

Thus $\mathrm d\alpha \ne 0$. The form $\alpha $ vanishes near the singular locus. Its operator norm relative to $H_0$ is bounded on its support, and its norm relative to $K_0$ is bounded everywhere. Choose $|\varepsilon |$ so small that both norms, multiplied by $|\varepsilon |$, are less than $1/2$. This proves positivity of $h$ in the ambient sense, semipositivity of $H$, and the inequality $K_0+\varepsilon \pi ^*\alpha \geq \tfrac 12K_0$.

The equation $\partial _ z\partial _{\bar z}v=\lambda -\rho $ on $E_1$, whose right-hand side has mean zero, has a smooth real periodic solution, obtained by dividing its nonzero Fourier coefficients. Its pullback $v_0$ satisfies $\mathrm{dd}^{\mathrm{c}}v_0=(\lambda -\rho )\kappa _1$, which proves the formula for $k$. The forms $H_0$ and $K_0$ are closed. Hence $\mathrm dk$ and $\mathrm d^{\mathrm c}k$ come from the complex surface $E_1\times E_2$, so that $\mathrm dk\wedge \mathrm d^{\mathrm c}k=0$ for degree reasons, and $\mathrm{dd}^{\mathrm{c}}k=0$. The same identities hold for $H$. The Leibniz rule now gives $\mathrm{dd}^{\mathrm{c}}(k^ r)=\mathrm{dd}^{\mathrm{c}}(H^ r)=0$ for all $r$; this uses both torsion identities, not only pluriclosedness. Expanding $(H+\mathrm{dd}^{\mathrm{c}}v_0)^ n$ and integrating by parts shows that every term containing $\mathrm{dd}^{\mathrm{c}}v_0$ integrates to zero. Hence the masses agree, and positivity of $k$ makes them positive. $\square$