Corollary 11.7.4.A97DA8CE2358 The function $\kappa _\omega $ is concave, positively homogeneous and invariant under the addition of $\mathrm{dd}^{\mathrm{c}}f$ for $f\in C^\infty (X,\mathbb R)$. Let $\theta $ and $\eta $ be smooth real $(1,1)$-forms and let $\varepsilon \geq 0$. If $-\varepsilon \omega \leq \theta -\eta \leq \varepsilon \omega $, then
\[ |\kappa _\omega (\theta )-\kappa _\omega (\eta )|\leq \varepsilon . \]
If smooth real $(1,1)$-forms $\theta _ j$ converge uniformly to a smooth form $\theta $ and $T_ j\in \mathscr C_\omega $ minimize their pairings, every weak cluster point of $(T_ j)$ minimizes the pairing with $\theta $.
Proof.
By (11.7.2), $\kappa _\omega $ is an infimum of linear functionals, which proves concavity and positive homogeneity. Invariance under the addition of $\mathrm{dd}^{\mathrm{c}}f$ follows from $\mathrm{dd}^{\mathrm{c}}T=0$. The normalization of the trace gives
\[ \left|\int _ X(\theta -\eta )\wedge T\right|\leq \varepsilon \quad (T\in \mathscr C_\omega ), \]
and taking infima proves the estimate. For the last assertion, pass to a subsequence $T_ j\rightharpoonup T\in \mathscr C_\omega $. The estimate and the uniform convergence of the backgrounds give
\[ \int _ X\theta \wedge T =\lim _ j\int _ X\theta _ j\wedge T_ j =\lim _ j\kappa _\omega (\theta _ j)=\kappa _\omega (\theta ). \]
Thus $T$ is a minimizer.
$\square$
Pome
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