  In 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.
