# reference for the weak compactness of currents

I am trying to follow the arguments in page 22 of the following paper k\"{a}hler currents and null loci

It quotes the weak compactness of currents, I wonder if there is any reference about it. My knowledge about currents are all from Demailly's notes

Basically, you can define a norm on the space of $$p$$ forms, and it will be a Frechet space, then you can talk about weak topology. Does weak compactness mean that the space of currents are weakly compact? If that is true, we also need a way to identify the space of $$p$$ forms as a subspace of the currents. I guess it can be done through the metric. Is there any reference that actually goes through all these constructions in detail?

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• The inclusion of smooth forms into currents is standard (example 2.5 chapter I page 15 of Demailly's book). The weak compactness statement is Proposition 1.23 in chapter III of Demailly's book (page 136). To apply it, take the function $\delta$ there to be a small positive constant, and note that the Kahler metrics $\omega(t)$ to which the proposition is applied live in a family of cohomology classes which lie inside a bounded set in cohomology, hence the integrals $\int_X \omega(t)\wedge\omega^{n-1}$ are bounded above by a fixed constant $C$ independent of $t$. Then take $\delta=1/C$. – YangMills Sep 12 at 14:15
• Thank you very much! – zach Sep 12 at 15:19