Arithmetic Bogomolov-gieseker’s Inequality

Let f : X → Spec(Z) be an arithmetic variety of dimension d ≥ 2 and (H, k) an arithmetically ample Hermitian line bundle on X, that is, a Hermitian line bundle with the following properties: (1) H is f -ample. (2) The Chern form c1(H∞, k) gives a Kähler form on X∞. (3) For every irreducible horizontal subvariety Y (i.e. Y is flat over Spec(Z)), the height ĉ1( (H, k)|Y ) dim Y of Y is positive. Let (E, h) be a rank r vector bundle on X. In this paper, we will prove that if E∞ is semistable with respect to H∞ on each connected component of X∞, then { ĉ2(E, h)− r − 1 2r ĉ1(E, h) 2 } · ĉ1(H, k) d−2 ≥ 0. Moreover, if the equality of the above inequality holds, then E∞ is projectively flat and h is a weakly Einstein-Hermitian metric.