Proposition 11.7.1155386134D9E

Proof. Gauduchon's theorem gives a smooth strictly positive member of $\mathscr G_\omega $ after normalization. The fixed trace mass bounds the total variation of all coefficient measures in finitely many local frames, so weak compactness of finite measures gives compactness of the family of positive currents with this trace. Positivity, the trace normalization and the equation $\mathrm{dd}^{\mathrm{c}}T=0$ pass to weak limits. On this bounded set, testing against smooth forms or against forms with continuous coefficients gives the same topology, by uniform approximation, and this topology is metrizable since the space of continuous test forms is separable. Thus the compactness can also be used sequentially. The defining conditions of $\mathscr G_\omega $ are convex. Its smooth strictly positive members form a convex set, so their closure $\mathscr C_\omega $ is compact and convex as well.

Pairing with a fixed smooth $\theta $ is weakly continuous. Its minimum on the closure of the smooth members is therefore their infimum, which proves (11.7.2). The Lamari criterion shows that (11.7.3) holds for every smooth strictly positive member, and hence for every member of their weak closure.

Conversely, suppose that $T\in \mathscr G_\omega $ does not belong to $\mathscr C_\omega $, and separate this point from the compact convex set $\mathscr C_\omega $ in the space of currents with the weak topology. A continuous linear functional for that topology is evaluation against a smooth form. After changing its sign if necessary, we obtain a smooth real $(1,1)$-form $\beta $ such that

By corollary 11.6.2, the smooth form $\beta -\kappa _\omega (\beta )\omega $ admits a positive current. Its pairing with $T$ is negative, because $\int _ X\omega \wedge T=1$. This contradicts (11.7.3) and proves the converse. $\square$