7.1. Basic properties [01GN]

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7.1. Basic properties

Fix a closed $(1,1)$ -form $\theta\in\mathcal{Z}^{1,1}(X)$ as above. From Proposition 5.8 and 5.9 we obtain:

Proposition 7.4.

The set $\PSH(X,\theta)$ is convex. If $\varphi$ is $\theta$ -psh and $c\in\mathbf{R}$ , then $\varphi+c$ is $\theta$ -psh. If $\theta$ is furthermore semipositive, then $\max\{\varphi,\psi\}$ is $\theta$ -psh when $\varphi,\psi\in\PSH(X,\theta)$ .

Proposition 7.5.

If $\varphi\in\PSH(X,\theta)$ and $\mathcal{X}$ is an SNC model on which $\theta$ is determined, then:

(i)

$\varphi\circ\emb_{\mathcal{X}}$ is continuous on $\Delta_{\mathcal{X}}$ and convex on each face;
(ii)

$\varphi\circ p_{\mathcal{X}}$ is continuous on $X$ and $\varphi\leq\varphi\circ p_{\mathcal{X}}$ .

The next result shows how to reconstruct a $\theta$ -psh function from its values on quasimonomial points.

Proposition 7.6.

Let $\varphi\in\PSH(X,\theta)$ . Then, as $\mathcal{X}$ runs through the directed set of SNC models on which $\theta$ is determined, $(\varphi\circ p_{\mathcal{X}})_{\mathcal{X}}$ forms a decreasing net of continuous functions on $X$ , converging pointwise to $\varphi$ .

Proof.

Let $\mathcal{X}^{\prime}\geq\mathcal{X}$ be two SNC models on which $\theta$ is determined. Then $p_{\mathcal{X}}\circ p_{\mathcal{X}^{\prime}}=p_{\mathcal{X}}$ . By Proposition 7.5 (ii) this implies $\varphi\leq\varphi\circ p_{\mathcal{X}^{\prime}}\leq\varphi\circ p_{\mathcal{X}}\circ p_{\mathcal{X}^{\prime}}=\varphi\circ p_{\mathcal{X}}$ , with equality on $\emb_{\mathcal{X}}(\Delta_{\mathcal{X}})$ . Set $\tilde{\varphi}:=\lim_{\mathcal{X}}\varphi\circ p_{\mathcal{X}}$ . Then $\tilde{\varphi}\geq\varphi$ . On the other hand, it follows from Corollary 3.2 that $p_{\mathcal{X}}$ converges to the identity on $X$ , so by upper semicontinuity of $\varphi$ we have $\varphi\geq\tilde{\varphi}$ . ∎