Arithmetic Demailly Approximation Theorem

We generalize the Demailly approximation theorem from complex geometry to Arakelov geometry.As an application, let $X\mathrm{/}\mathbb{Q}$ be an integral projective variety and $\bar{N}$ be an adelic line bundle on $X$, we prove that $\mathrm{ess}(\bar{N})\ge 0$ $⟹$ $\bar{N}$ pseudo-effective. This was proved in [Bal21], 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 [YZ22].