5.1. Positive closed ( 1 , 1 ) -forms and metrics [01FF]

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5.1. Positive closed $(1,1)$ -forms and metrics

The following definition extends the one in [Zha95, Gub98, CL06].

Definition 5.1.

A closed $(1,1)$ -form $\theta$ is said to be:

(i)

semipositive if $\theta_{\mathcal{X}}\in N^{1}(\mathcal{X}/S)$ is nef for some (or, equivalently, any) determination $\mathcal{X}$ of $\theta$ ;
(ii)

$\mathcal{X}$ -positive if $\mathcal{X}\in\mathcal{M}_{X}$ is a determination of $\theta$ and $\theta_{\mathcal{X}}\in N^{1}(\mathcal{X}/S)$ is ample.

A model metric $\|\cdot\|$ on a line bundle $L$ is said to be semipositive if the curvature form $c_{1}(L,\|\cdot\|)$ is semipositive.

The equivalence in (i) follows from the following standard fact: if $\alpha\in N^{1}(\mathcal{X}/S)$ is a numerical class and $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ is a vertical blow-up then $\pi^{*}\alpha$ is nef iff $\alpha$ is nef. On the other hand, the analogous result is obviously wrong for ample classes, so that it is indeed necessary to specify the model in (ii). If $\omega$ is $\mathcal{X}$ -positive and $\theta\in\mathcal{Z}^{1,1}(X)$ is determined on $\mathcal{X}$ then $\omega+\varepsilon\theta$ is also $\mathcal{X}$ -positive for all $0<\varepsilon\ll 1$ .

The set of all semipositive closed $(1,1)$ -forms is a convex cone $\mathcal{Z}^{1,1}_{+}(X)$ of $\mathcal{Z}^{1,1}(X)$ that can be equivalently defined as

\mathcal{Z}^{1,1}_{+}(X):=\varinjlim_{\mathcal{X}}\Nef(\mathcal{X}/S).

Proposition 5.2.

Let $\theta$ be a closed $(1,1)$ -form whose de Rham class $\{\theta\}\in N^{1}(X)$ is ample. For every sufficiently high model $\mathcal{X}$ , we may then find a model function $\varphi$ such that $\theta+dd^{c}\varphi$ is $\mathcal{X}$ -positive. If $\theta$ is furthermore semipositive then we may also arrange that $-\varepsilon\leq\varphi\leq 0$ for any given $\varepsilon>0$ .

Proof.

Let $\mathcal{X}^{\prime}$ be a determination of $\theta$ and let $\mathcal{L}^{\prime}\in\Pic(\mathcal{X}^{\prime})_{\mathbf{R}}$ be a representative of $\theta$ . The assumption implies that the $\mathbf{R}$ -line bundle $L:=\mathcal{L}^{\prime}|_{\mathcal{X}_{K}}$ is ample. By Corollary 1.5 we may thus assume that $\mathcal{X}^{\prime}$ has been chosen so that $L$ admits an ample extension $\mathcal{L}\in\Pic(\mathcal{X})_{\mathbf{R}}$ for each model $\mathcal{X}$ dominating $\mathcal{X}^{\prime}$ . If $\pi:\mathcal{X}\to\mathcal{X}^{\prime}$ denotes the corresponding vertical blow-up then $\mathcal{L}-\pi^{*}\mathcal{L}^{\prime}=D$ for some $D\in\Div_{0}(\mathcal{X})_{\mathbf{R}}$ , and $\varphi=\varphi_{D}$ is a model function such that $\theta+dd^{c}\varphi$ is $\mathcal{X}$ -positive.

Now suppose $\theta$ is semipositive and pick $\mathcal{X}$ , $\varphi$ as above. Upon replacing $\varphi$ by $\varphi-\sup_{X}\varphi$ we may assume that $\varphi\leq 0$ . Then the closed $(1,1)$ -form

\theta+dd^{c}(\varepsilon\varphi)=\varepsilon(\theta+dd^{c}\varphi)+(1-\varepsilon)\theta

is also $\mathcal{X}$ -positive for each $0<\varepsilon<1$ , completing the proof since $\varphi$ is bounded. ∎

Since the nef cone of $N^{1}(X)$ is the closure of the ample cone, we get as a consequence:

Corollary 5.3.

The closure of the image of $\mathcal{Z}_{+}^{1,1}(X)$ in $N^{1}(X)$ coincides with the nef cone of $N^{1}(X)$ .

Remark 5.4.

In the complex case, it is not always possible to find a smooth semipositive form in a nef class, so the image of $\mathcal{Z}^{1,1}_{+}(X)$ in $N^{1}(X)$ is strictly contained in $\Nef(X)$ in general, see [DPS94, Example 1.7]. In the non-Archimedean setting, the situation is unclear.