Arithmetic Demailly Approximation Theorem

Abstract

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