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Corollary 12.34.790A4213A9E87

Corollary 12.34.7.90A4213A9E87 Let

\[ b=\max _{T\in \mathscr G_{\omega _0}}\int _ X\chi \wedge T. \]

Then the maximal smooth existence time is

\[ \tau =\begin{cases} 1/b,& b>0,\\ +\infty ,& b\leq 0.\end{cases} \]

If $\tau <\infty $, the limiting background $\theta _\tau $ is nef and there is a current $T_*\in \mathscr G_{\omega _0}$ with $\int _ X\theta _\tau \wedge T_*=0$. If moreover $\lambda _\mu (\theta _\tau )>0$, every such annihilating current has zero determinant-weighted measure.

Proof. For $t\geq 0$, the normalization of the test currents gives the identity

\[ m_{\omega _0}(\theta _ t)=1-tb. \]

Theorem 12.34.1, Theorem 12.34.4 give the formula for $\tau $. When $b>0$, the maximum defining $b$ is attained by weak compactness, and a maximizer $T_*$ annihilates $\theta _\tau $. The same identity gives $m_{\omega _0}(\theta _\tau )=0$, hence nefness. The last assertion is proposition 12.34.2. $\square$

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