Proposition 12.6.4367B12ACF7773

Proof. Put $L=ca+e$, and suppose that a quasi-plurisubharmonic $u$ satisfies $\theta +\mathrm{dd}^{\mathrm{c}}u\geq \delta h$ for some $\delta \geq 0$. Pairing with the closed semipositive form $\kappa ^{m-1}\wedge w$ and using Stokes' theorem for currents gives $b\geq \delta $. For every smooth nonnegative top-degree form $\eta $ on $B$, its pullback to $X$ is closed for degree reasons, and

If $L<\delta $ at some point, a nonzero form $\eta $ supported in the open set where $L<\delta $ contradicts this inequality. Hence $L\geq \delta $ everywhere. For $\delta =0$ these conditions are exactly $b\geq 0$, $e\geq 0$ and $e+cM\geq 0$; conversely, they make $\theta $ semipositive, and such forms are nef because $\theta +\varepsilon h>0$ for every $\varepsilon >0$. Conversely, if $\theta $ is nef, applying the same tests to smooth positive representatives of $\theta +\varepsilon h$ and letting $\varepsilon $ tend to zero gives these inequalities. For $\delta >0$ the conditions force the three strict inequalities, and conversely these make $\theta $ itself strictly positive. $\square$