Arithmetic Demailly approximation theorem

We generalize the Demailly approximation theorem from complex geometry to Arakelov geometry.

As an application, let $X/Q$ be an integral projective variety and $\bar{N}$ be an adelic line bundle on X. We prove that $\mathrm{ess}\left(\bar{N}\right)\ge 0$ ⟹ $\bar{N}$ pseudo-effective. This was proved in [1], assuming $\bar{N}$ relatively semipositive.

We show in the appendix that the above assertion is also true for adelic line bundles on quasi-projective varieties, under the framework of [17].