  Let K be an algebraic number field, O_K the ring of integers of K, and f : X
--> Spec(O_K) an arithmetic surface. Let (E, h) be a rank r Hermitian vector
bundle on X such that $E$ is semistable on the geometric generic fiber of f. In
this paper, we will prove an arithmetic analogy of Bogomolov-Gieseker's
inequality: c_2(E, h) - (r-1)/(2r) c_1(E, h)^2 >= 0.
