In this paper, we will consider a generalization of Bogomolov’s inequality and Cornalba-Harris-Bost’s inequality to the case of 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 in [22]. We also establish the arithmetic Riemann-Roch formulae for stable curves over regular arithmetic varieties and generically finite morphisms of arithmetic varieties.