Arithmetic Demailly Approximation Theorem

TL;DR

This paper extends Demailly's approximation theorem from complex to Arakelov geometry and demonstrates that certain adelic line bundles are pseudo-effective when their essential minimum is non-negative.

Contribution

It generalizes the Demailly approximation theorem to Arakelov geometry and establishes pseudo-effectiveness of adelic line bundles under broader conditions.

Findings

Generalization of Demailly approximation theorem to Arakelov geometry

Proved pseudo-effectiveness of adelic line bundles with non-negative essential minimum

Extended results to quasi-projective varieties in the appendix

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 $\operatorname{ess}(\overline N) \geq 0$ $\Longrightarrow $ $\overline N$ 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].