12.6.42 Current cones and endpoint equations9FC10688E696

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. $\square$

Proof. Let $f=a$ for $\theta _+(b)$ and $f=1-a/M$ for $\theta _-(b)$, and let $r>1$ be the corresponding ratio $r_0$ or $r_ M$. The coefficient $L_0$ is a positive constant multiple of $f$. In each tube around an extreme component, the integral of $f^{-p}$ converges exactly when

converges, where $s$ is the normal radius and $s_0>0$ is small. Thus $1/L_0\in L^ p(B,\sigma )$ for every $1<p<r$. The zero set of $L_0$ has volume zero, so the value assigned to the density there is irrelevant. The formulas for $\theta _\pm (b)$ give

We apply theorem 14.19.2 on $B$ to the positive closed background $b\kappa $ and obtain a continuous $\phi $ with $A=b\kappa +\mathrm{dd}^{\mathrm{c}}\phi \geq 0$ and $A^ m=c_0\sigma /L_0$ for a constant $c_0>0$. Since $b\kappa $ is closed, bounded potentials do not change its mass, by theorem 11.1.7; hence $\int _ BA^ m=b^ m$ and $c_0=b^ m/H_0$. Set $u=\phi \circ p_ B$, where $p_ B:X\to B$ is the projection. Then $\theta +\mathrm{dd}^{\mathrm{c}}u=A+L_0w\geq 0$.

The pullback identity of lemma 5.28.4 gives $A^{m+1}=0$ on $X$, and $(L_0w)^2=0$. The multilinear expansion of proposition 5.28.2 therefore gives

The cancellation $L_0(\sigma /L_0)=\sigma $ holds as measures, since the zero set of $L_0$ has volume zero and $A^ m$ is absolutely continuous with density $c_0/L_0$ relative to $\sigma $.

Suppose that some solution $q$, not necessarily pulled back from $B$, were of class $C^{1,1}$ on $X$. Since $\mu /h^ n$ is smooth and strictly positive on compact $X$, proposition 5.28.6 would give $\theta +\mathrm{dd}^{\mathrm{c}}q\geq \delta h$ for some $\delta >0$. This contradicts proposition 12.6.43, because $L_0$ vanishes somewhere. $\square$