Proposition 10.11.2A8D846A01D78

Proof. The finite-dimensional space of smooth representatives in lemma 20.4.20 identifies Bott–Chern cohomology with a finite-dimensional vector space and gives it its usual topology; the trace functional is continuous for this topology. Its positivity and the absence of lines in the cone were proved in subsection 20.4.9, and convexity follows by adding positive currents.

Let pseudoeffective classes $\alpha _ j$ converge to $\alpha $, and choose positive closed representatives $T_ j$. Their masses $\ell _\gamma (\alpha _ j)$ are bounded, so by weak compactness a subsequence converges to a positive closed current $T$. Let $B_\gamma $ be the continuous smooth representative map of lemma 20.4.20. Along the subsequence, $B_\gamma (T_ j)$ converges both to $B_\gamma (T)$ and to the representative of $\alpha $, so $T$ represents $\alpha $. Thus the cone is closed. On its mass-one slice, (20.4.21) bounds $B_\gamma (T)$ in the finite-dimensional representative space. The slice is therefore closed and bounded there, hence compact.

The class maps of a modification are linear, and they are well-defined because classes can be represented equally by smooth forms and by currents. As linear maps between finite-dimensional spaces, they are continuous. Pullback preserves positive closed currents, since it can be computed on local plurisubharmonic potentials as in proposition 20.4.22, and proper pushforward preserves positivity and closedness. Applying these maps to positive representatives proves the assertions about the cones. $\square$