Lemma 22.1.10

Lemma 22.1.10.9A2678C2C223 The image of the canonical injection

\[ \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}\hookrightarrow \mathrm{PSH}^{\mathrm{NA}}(X,\theta ) \]

is given by the set of $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ with positive volume.

Proof. By proposition 22.1.8, every element in the image has positive volume.

Conversely, take $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ with positive volume. We want to construct $\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ representing $\Gamma $.

Fix a Kähler form $\omega $ on $X$. Define

717C364BF812 22.1.11

\begin{equation} \Gamma '_{\tau }\coloneqq \lim _{k\to \infty }\Gamma ^{\theta +k^{-1}\omega }_{\tau },\quad \tau <\Gamma _{\max }. \end{equation}

We claim that it suffices to show

80067294986D 22.1.12

\begin{equation} \varliminf _{k\to \infty }\int _ X \left(\theta +k^{-1}\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +k^{-1}\omega }_{\tau }\right)^ n>0 \end{equation}

for some $\tau <\Gamma _{\max }$. If this holds, then the argument of 17.12.6 implies that the same holds for all $\tau <\Gamma _{\max }$. Then proposition 16.4.4, proposition 16.6.29 guarantee that $\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ and represents $\Gamma $.

It remains to argue (22.1.12). Let $\epsilon =\operatorname{vol}\Gamma >0$. Take $\tau <\Gamma _{\max }$ so that

\[ \int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{-\infty }\right)^ n-\int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{\tau }\right)^ n<\epsilon /2. \]

Therefore, by 22.1.9, for any $k\geq 1$,

\[ \int _ X \left(\theta +k^{-1}\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +k^{-1}\omega }_{-\infty }\right)^ n-\int _ X \left(\theta +k^{-1}\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +k^{-1}\omega }_{\tau }\right)^ n<\epsilon /2. \]

By the definition of $\operatorname{vol}\Gamma $, the first integral in the preceding line tends to $\epsilon $ as $k\to \infty $. Hence

\[ \varliminf _{k\to \infty }\int _ X \left(\theta +k^{-1}\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +k^{-1}\omega }_{\tau }\right)^ n\geq \epsilon /2>0, \]

which proves (22.1.12). $\square$

Lemma 22.1.10`9A2678C2C223`

Comments

Lemma 22.1.109A2678C2C223

Comments

Lemma 22.1.10`9A2678C2C223`