Arithmetic Fujita approximation over adelic curves
Chunhui Liu
TL;DR
The work develops arithmetic Fujita approximation within the framework of Arakelov theory on adelic curves, proving that arithmetically big $\mathbb{R}$-line bundles on a projective variety over a proper adelic curve can be birationally decomposed after a suitable power so that an arithmetically ample factor captures most of the volume up to $\varepsilon$. The approach builds a cohesive infrastructure: defining adelic curves and vector bundles, equipping them with Harder–Narasimhan filtrations and induced measures, and then transferring these filtrations to graded linear series (including those over trivially valued fields) to obtain approximability and convergence of volume-related invariants. The core contributions include (i) a graded-linear-series Fujita approximation in the adelic setting, (ii) a full arithmetic Fujita approximation for big adelic line bundles over adelic curves, and (iii) a detailed treatment of $\mathbb{Q}$- and $\mathbb{R}$-line bundles via density arguments, with explicit volume inequalities. These results unify and extend previous arithmetic Fujita results over number fields to the broader adelic-curve framework, enabling robust control of volumes and positivity in arithmetic geometry.
Abstract
In this paper, we will prove an analogue of Fujita's approximation theorem under the framework of Arakelov theory over adelic curves, which proves a conjecture of Huayi Chen and Atsushi Moriwaki.