2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/225In this paper, we will consider a generalization of Bogomolov's inequality and Cornalba-Harris-Bost's inequality to semistable families of arithmetic varieties under the idea that geometric semistability implies a certain kind of arithmetic positivity. The first one is an arithmetic analogue of the relative Bogomolov's inequality proved by the second author. We also establish the arithmetic Riemann-Roch formulae for stable curves over regular arithmetic varieties and generically finite morphisms of arithmetic varieties.Version 3.0 (75 pages), the new version of the paper titled "Relative Bogomolov's inequality in the arithmetic case"Algebraic Geometrytext