Proposition 10.4.23B6E3B145113

Proof. Sums and positive multiples of positive currents, Kähler forms or nef representatives are of the same kind, which proves convexity of all four cones.

Let $\alpha _ j\to \alpha $ be pseudo-effective classes, with positive currents $T_ j\in \alpha _ j$. Since the $\mathrm{dd}^{\mathrm{c}}$ of a distribution pairs to zero with the closed form $\omega ^{n-1}$, the trace masses are

where $\theta _{\alpha _ j}$ are the representatives of (10.2.18); they are bounded because the classes converge. Let $K$ be a compact subset of a chart, $\chi $ a test function on the chart with $0\leq \chi \leq 1$ and $\chi =1$ on $K$, and $c'>0$ such that $\beta \leq c'\omega $ on $\operatorname{Supp}\chi $ (lemma 6.14.1), where $\beta $ is the Euclidean form (5.2.8) of the chart. By (10.2.10), applied with $\gamma _ j=c'\omega $ and $\gamma '_ j=\beta $, the form $(c'\omega )^{n-1}-\beta ^{n-1}$ is a sum of products of semipositive $(1,1)$-forms on $\operatorname{Supp}\chi $, hence strongly positive (proposition 5.3.2). Therefore $c'^{n-1}\omega ^{n-1}-\chi \beta ^{n-1}=(1-\chi )c'^{n-1}\omega ^{n-1}+\chi \bigl (c'^{n-1}\omega ^{n-1}-\beta ^{n-1}\bigr )$, with $\chi \beta ^{n-1}$ extended by zero outside the chart, is a smooth strongly positive form on $X$, so the trace measures satisfy $\mu _{T_ j}(K)\leq T_ j(\chi \beta ^{n-1})\leq c'^{n-1}\int _ XT_ j\wedge \omega ^{n-1}$, which is bounded independently of $j$. By the compactness assertion of theorem 5.3.7, a subsequence of $(T_ j)$ converges weakly to a closed positive current $T$, and $\{ T\} =\alpha $ because the class map is weakly continuous. Hence $\mathcal E$ is closed.

If $\omega _0\in \alpha $ is a Kähler form, then $\omega _0\geq c\omega $ for some $c>0$, and by (10.2.18) the form $\omega _0+\theta _{\alpha '-\alpha }\in \alpha '$ is Kähler whenever $C\| \alpha '-\alpha \| <c$; hence $\mathcal K$ is open. If $\alpha $ is nef and $\varepsilon >0$, a representative $\theta _\varepsilon \geq -\varepsilon \omega $ gives the Kähler form $\theta _\varepsilon +2\varepsilon \omega \in \alpha +2\varepsilon \{ \omega \} $, so $\alpha \in \overline{\mathcal K}$. Conversely, let Kähler classes $\alpha _ j$ with Kähler forms $\omega _ j\in \alpha _ j$ converge to $\alpha $. Then $\omega _ j+\theta _{\alpha -\alpha _ j}\in \alpha $ is at least $-C\| \alpha -\alpha _ j\| \omega $, so $\alpha $ is nef; thus $\mathcal N=\overline{\mathcal K}$, and $\mathcal N\subset \mathcal E$ because $\mathcal K\subset \mathcal E$ and $\mathcal E$ is closed.

Since $\mathcal K$ is open and contained in $\mathcal N$, it lies in the interior of $\mathcal N$. If $\alpha $ is nef and $t>0$, then $\alpha +t\{ \omega \} $ is Kähler: take a representative $\theta _\varepsilon \geq -\varepsilon \omega $ with $\varepsilon <t$. If $\alpha $ lies in the interior of $\mathcal N$, then $\alpha -t\{ \omega \} $ is nef for small $t>0$, and therefore $\alpha =(\alpha -t\{ \omega \} )+t\{ \omega \} $ is Kähler. Thus $\mathcal K$ is the interior of $\mathcal N$.

If $T\in \alpha $ and $T\geq \delta \omega $, then $T+\theta _{\alpha '-\alpha }\in \alpha '$ is positive whenever $C\| \alpha '-\alpha \| \leq \delta $, so a neighborhood of $\alpha $ lies in $\mathcal E$. Conversely, if $\alpha $ lies in the interior of $\mathcal E$, then $\alpha -\delta \{ \omega \} \in \mathcal E$ for some $\delta >0$, and adding $\delta \omega $ to a positive current in $\alpha -\delta \{ \omega \} $ gives a Kähler current in $\alpha $. Hence $\mathcal B$ is the interior of $\mathcal E$. $\square$