2.4. θ -psh functions [0192]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

2.4. $\theta$ -psh functions

Fix a form $\theta\in\mathcal{Z}^{1,1}(X)$ with ample de Rham class $\{\theta\}\in N^{1}(X)$ .

Definition 2.5.

A $\theta$ -psh function $\varphi:X\to[-\infty,+\infty[$ is an usc function such that for each SNC model $\mathcal{X}$ of $X$ on which $\theta$ is determined we have

(i)

$\varphi\leq\varphi\circ p_{\mathcal{X}}$ on $X$ ;
(ii)

the restriction of $\varphi$ to the dual complex $\Delta_{\mathcal{X}}$ is a uniform limit of restrictions of model functions $\psi$ such that $\theta+dd^{c}\psi$ is a semipositive form.

We write $\PSH(X,\theta)$ for the set of $\theta$ -psh functions on $X$ .

It is a nontrivial fact that if $\varphi$ is a $\theta$ -psh model function then the form $\theta+dd^{c}\varphi$ is in fact semipositive, see [BFJ11, Theorem 5.11]. In particular, the zero function is $\theta$ -psh iff $\theta$ is semipositive. In this case, $\max\{\varphi,-t\}$ is $\theta$ -psh when $\varphi$ is $\theta$ -psh and $t\in\mathbf{R}$ .

Proposition 2.6.

[BFJ11, Proposition 5.10]. The space of model functions $\mathcal{D}(X)$ is spanned by $\theta$ -psh model functions.

Proposition 2.7.

[BFJ11, Proposition 7.4]. The set $\PSH(X,\theta)$ is convex. If $\varphi,\psi$ are $\theta$ -psh and $c\in\mathbf{R}$ , then the functions $\max\{\varphi,\psi\}$ and $\varphi+c$ are also $\theta$ -psh.

Proposition 2.8.

[BFJ11, Proposition 7.5]. Any $\varphi\in\PSH(X,\theta)$ is continuous on the dual complex of any SNC model $\mathcal{X}$ , and convex on each of its faces.

In fact, the continuity statement above can be made uniform in $\varphi$ :

Theorem 2.9.

[BFJ11, Corollary 7.7] For any SNC model $\mathcal{X}$ , the restrictions of all $\theta$ -psh functions to the dual complex $\Delta_{\mathcal{X}}$ form an equicontinuous family.

We endow $\PSH(X,\theta)$ with the topology of uniform convergence on dual complexes. Notice that the divisorial points are dense on each dual complex $\Delta_{X}$ , see [BFJ11, Corollary 3.13] or [JM10, Remark 3.9]. As a consequence of equicontinuity we thus have

Theorem 2.10.

[BFJ11, Theorem 7.8]. For each model function $\psi$ the map $\varphi\mapsto\sup_{X}(\varphi-\psi)$ is continuous and proper on $\PSH(X,\theta)$ . In particular, the space $\PSH(X,\theta)/\mathbf{R}$ is compact. Further, the topology on $\PSH(X,\theta)$ is equivalent to the topology of pointwise convergence on $X^{\mathrm{div}}$ .

Finally we have the following regularization result. Its proof relies on multiplier ideals.

Theorem 2.11.

[BFJ11, Theorem 8.7]. For any $\theta$ -psh function $\varphi$ , there exists a decreasing net $(\varphi_{j})_{j}$ of $\theta$ -psh model functions that converges pointwise on $X$ to $\varphi$ .

The complex analogue of this result is due to Demailly [Dem92] (see also [GZ05, Appendix] for the case of a line bundle). By Dini’s lemma, we get as a consequence:

Corollary 2.12.

[BFJ11, Corollary 8.8] The set $\mathcal{D}(X)\cap\PSH(X,\theta)$ is dense in $C^{0}(X)\cap\PSH(X,\theta)$ with respect to uniform convergence on $X$ .

Proposition 4.5 below refines Theorem 2.11 and asserts that any $\theta$ -psh function is actually the decreasing limit of a sequence of $\theta$ -psh model functions (but the proof heavily uses Theorem 2.11).