8.3. Continuity [01C3]

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8.3. Continuity

Finally we show that $\varphi$ is continuous. For this we use capacity estimates in the spirit of Kołodziej [Koł98, Koł03]; see also [EGZ09]. The following result (and its proof) is a translation of [EGZ09, Lemma 2.3].

Lemma 8.3.

Let $\varphi,\psi\in\mathcal{E}^{1}(X,\omega)$ with $\psi\leq 0$ . Then

\Capa_{\omega}\{\varphi<\psi\}\leq t^{-n}\int_{\left\{\varphi<(1-t)\psi+t\right\}}\MA(\varphi)

for $0<t<1$ .

Proof.

Fix $u\in\PSH(X,\omega)$ with $0\leq u\leq 1$ and set $\psi_{t}:=(1-t)\psi+tu$ . We have

\{\varphi<\psi\}\subseteq\{\varphi<\psi_{t}\}\subseteq\{\varphi<(1-t)\psi+t\}.

since $\psi\leq 0$ . Now $\MA(\psi_{t})\geq t^{n}\MA(u)$ by (3.1), so

t^{n}\int_{\{\varphi<\psi\}}\MA(u)\leq\int_{\{\varphi<\psi\}}\MA(\psi_{t})\leq\int_{\{\varphi<\psi_{t}\}}\MA(\psi_{t})\\ \leq\int_{\{\varphi<\psi_{t}\}}\MA(\varphi)\leq\int_{\{\varphi<(1-t)\psi+t\}}\MA(\varphi),

where the third inequality follows from the comparison principle (6.7). Taking the supremum over $u$ completes the proof. ∎

As a consequence, we get the following version of the ’domination principle’, sufficient for our purpose.

Lemma 8.4.

Let $\varphi\in\PSH(X,\omega)\cap C^{0}(X)$ and $\psi\in\mathcal{E}^{1}(X,\omega)$ . Assume that $\nu:=\MA(\psi)$ is supported in the dual complex $\Delta_{\mathcal{X}}$ of some SNC model $\mathcal{X}$ , and that $\varphi\leq\psi$ $\nu$ -a.e. Then $\varphi\leq\psi$ on $X$ .

Proof.

Upon translating by a constant we may assume that $0\geq\varphi\geq-C$ . Let $\varepsilon>0$ . If we choose $0<t\ll 1$ such that $t(C+1)\leq\varepsilon/2$ then we have

\nu\{\psi+\varepsilon<(1-t)\varphi+t\}\leq\nu\{\psi+\varepsilon/2<\varphi\}=0.

By Lemma 8.3 it follows that

\Capa_{\omega}\{\psi+\varepsilon<\varphi\}\leq t^{-n}\nu\{\psi+\varepsilon<(1-t)\varphi+t\}=0

(since $\MA(\psi+\varepsilon)=\nu$ ). But $\{\psi+\varepsilon<\varphi\}$ is open by continuity of $\varphi$ , hence empty by Lemma 4.2. We have thus proved that $\varphi\leq\psi+\varepsilon$ on $X$ for all $\varepsilon>0$ , and the result follows. ∎

Now let $\varphi\in\mathcal{E}^{1}(X,\omega)$ be a solution to $\MA(\varphi)=\mu$ , with $\mu$ supported in a dual complex $\Delta_{\mathcal{X}}$ . We may normalize $\varphi$ by $\sup_{X}\varphi=-1$ . Let $(\varphi_{j})_{j}$ be a decreasing net of $\omega$ -psh model functions converging to $\varphi$ . We are going to show that $\varphi_{j}\to\varphi$ uniformly on $X$ , which will in particular imply that $\varphi$ is continuous.

By Theorem 2.10 we have $\sup_{X}\varphi_{j}\to\sup_{X}\varphi$ , so we may assume $\varphi_{j}\leq 0$ for all $j$ . Fix $\varepsilon>0$ . Since $\varphi$ is continuous on $\Delta_{\mathcal{X}}$ , the monotone convergence $\varphi_{j}\to\varphi$ is uniform on $\Delta_{\mathcal{X}}$ by Dini’s lemma. We thus have $\varphi_{j}\leq\varphi+\varepsilon$ $\mu$ -a.e. for $j\gg 1$ , and Lemma 8.4 yields $\varphi_{j}\leq\varphi+\varepsilon$ on $X$ , which concludes the proof.