7.4. Upper envelopes [01GY]

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7.4. Upper envelopes

As a consequence of compactness we shall prove the following result, whose complex analogue serves as a basic ingredient of pluripotential theory. While we will not go deeper into pluripotential theory here, we will use the result below in §8.

Theorem 7.9.

Let $(\varphi_{\alpha})_{\alpha\in A}$ be an arbitrary set of $\theta$ -psh functions on $X$ and assume that $(\varphi_{\alpha})$ is uniformly bounded from above. If we set $\varphi(x):=\sup_{\alpha\in A}\varphi_{\alpha}(x)$ for each $x\in X$ , then the usc regularization $\varphi^{*}$ of $\varphi$ is $\theta$ -psh and coincides with $\varphi$ on $X^{\mathrm{qm}}=\bigcup_{\mathcal{X}}\emb_{\mathcal{X}}(\Delta_{\mathcal{X}})$ .

Recall that the usc regularization of a function $u$ on a topological space $X$ is the smallest usc function $u^{*}\geq u$ .

Lemma 7.10.

Let $u:X\to[-\infty,+\infty[$ be a function such that for each SNC model $\mathcal{X}$ we have

(i)

$u\circ i_{\mathcal{X}}$ is continuous on $\Delta_{\mathcal{X}}$ .
(ii)

$u\leq u\circ p_{\mathcal{X}}$ .

Then $u^{*}=\inf_{\mathcal{X}}u\circ p_{\mathcal{X}}$ , hence $u^{*}\circ i_{\mathcal{X}}=u\circ i_{\mathcal{X}}$ for all $\mathcal{X}$ .

Proof.

Condition (i) implies that $u\circ p_{\mathcal{X}}=u\circ i_{\mathcal{X}}\circ p_{\mathcal{X}}$ is continuous for all $\mathcal{X}$ , so that $v:=\inf_{\mathcal{X}}u\circ p_{\mathcal{X}}$ is usc. It follows that $v\geq u^{*}$ , since $v\geq u$ by (ii). Conversely, for each $x\in X$ we have $\lim_{\mathcal{X}}p\mathcal{X}(x)=x$ , hence

u^{*}(x)\geq\limsup_{\mathcal{X}}u^{*}\circ p_{\mathcal{X}}(x)\geq\inf_{\mathcal{X}}u\circ p_{\mathcal{X}}(x)=v(x),

which shows that $v=u^{*}$ . Finally, (ii) shows that $\mathcal{X}^{\prime}\geq\mathcal{X}\Rightarrow u\circ p_{\mathcal{X}^{\prime}}\leq u\circ p_{\mathcal{X}}$ , hence $v\circ p_{\mathcal{X}}=u\circ p_{\mathcal{X}}$ , which is equivalent to the last assertion. ∎

Proof of Theorem 7.9.

Upon considering the new family $\varphi_{I}=\max_{\alpha\in I}\varphi_{\alpha}$ with $I$ ranging over all finite subsets of $A$ , we may assume that $A$ is a directed set and $(\varphi_{\alpha})$ is an increasing net. For each SNC model $\mathcal{X}$ we have $\varphi_{\alpha}\leq\varphi_{\alpha}\circ p_{\mathcal{X}}$ for all $\alpha$ , hence $\varphi\leq\varphi\circ p_{\mathcal{X}}$ . By Corollary 7.7 $\varphi_{\alpha}\circ i\mathcal{X}$ converges uniformly to $\varphi\circ i_{\mathcal{X}}$ , which is therefore continuous. Using Lemma 7.10 we conclude that $\varphi^{*}$ is usc, satisfies $\varphi^{*}\leq\varphi^{*}\circ p_{\mathcal{X}}$ , and $\varphi^{*}\circ i_{\mathcal{X}}=\varphi\circ i_{\mathcal{X}}$ is a uniform limit of restrictions to $\Delta_{\mathcal{X}}$ of $\theta$ -psh functions, hence $\varphi^{*}$ is $\theta$ -psh. ∎