Pome

Lemma 19.4.44F3D0645BDF93

Lemma 19.4.44.F3D0645BDF93 Assume that $\theta _{\varphi }$ is a Kähler current. Then as $\mathbb {Q}\ni \beta \to 0+$, we have

\[ \Delta _{\nu }\Bigl ((1-\beta )\theta ,\varphi \Bigr )\xrightarrow {d_{\mathrm{Haus}}} \Delta _{\nu }(\theta ,\varphi ). \]

Proof. By proposition 19.4.33 and proposition 19.4.28, we have

\[ \Delta _{\nu }\Bigl ((1-\beta )\theta ,\varphi \Bigr )+\beta \Delta _{\nu }(L)\subseteq \Delta _\nu \left(\theta ,\varphi +\beta V_\theta \right)\subseteq \Delta _{\nu }(\theta ,\varphi ) \]

for any small enough $\beta \in \mathbb {Q}_{>0}$. In particular, if $\Delta '$ is the Hausdorff limit of a subnet of $(\Delta ((1-\beta )\theta ,\varphi ))_{\beta }$, then $\Delta '\subseteq \Delta _{\nu }(\theta ,\varphi )$. But

\[ \begin{split} \operatorname{vol}\Delta '=& \lim _{\beta \to 0+}\operatorname{vol}\Delta _{\nu }((1-\beta )\theta ,\varphi )\\ =& \frac{1}{n!}\lim _{\beta \to 0+}\int _ X \left((1-\beta )\theta +\mathrm{dd}^{\mathrm{c}}P_{(1-\beta )\theta }[\varphi ]_{\mathcal{I}}\right)^ n\\ =& \frac{1}{n!}\int _ X (\theta +\mathrm{dd}^{\mathrm{c}}P_{\theta }[\varphi ]_{\mathcal{I}})^ n. \end{split} \]

Since we have not developed the theory of nef b-divisors yet, we give a direct proof as well. Take a Kähler form $\omega $ so that $\theta _{\varphi }\geq \omega $. Let $\psi =P_{\theta -\omega }[\varphi ]_{\mathcal{I}}$. Then $\varphi \sim _{\mathcal{I}}\psi $. In order to establish the last equality, we may replace $\varphi $ by $\psi $ and hence assume that $\varphi $ is $\mathcal{I}$-good. For small enough $\beta >0$, the form $(1-\beta )\theta -(\theta -\omega )=\omega -\beta \theta $ is Kähler, so this remains true in the class $(1-\beta )\{ \theta \} $ by 18.1.8. Thus the equality of masses in an $\mathcal{I}$-equivalence class reduces the desired equality to

\[ \lim _{\beta \to 0+}\int _ X ((1-\beta )\theta +\mathrm{dd}^{\mathrm{c}}\varphi )^ n=\int _ X \theta _{\varphi }^ n, \]

which is obvious.

It follows that $\Delta '=\Delta _{\nu }(\theta ,\varphi )$. We conclude by theorem 19.7.4. $\square$

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