Inequalities for Semistable Families of Arithmetic Varieties

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. Introduction In this paper, we will consider a generalization of Bogomolov’s inequality and CornalbaHarris-Bost’s inequality to the case of semistable families of arithmetic varieties. An underlying idea of these inequalities as in [4], [5], [8], [17], [18], [19], [20], [21], [24], and [27] is that geometric semistability implies a certain kind of arithmetic positivity. The first one is related to the semistability of vector bundles, and the second one involves the Chow (or Hilbert) semistability of cycles. First of all, let us consider Bogomolov’s inequality. Let X and Y be smooth algebraic varieties over an algebraically closed field of characteristic zero, and f : X → Y a semistable curve. Let E be a vector bundle of rank r on X, and y a point of Y . In [22], the second author proved that if f is smooth over y and E|Xȳ is semistable, then disX/Y (E) = f∗ (2rc2(E)− (r − 1)c1(E)) is weakly positive at y. In the first half of this paper, we would like to consider an arithmetic analogue of the above result. Let us fix regular arithmetic varieties X and Y , and a semistable curve f : X → Y . Since we have a good dictionary for translation from a geometric case to an arithmetic case, it looks like routine works. There are, however, two technical difficulties to work over the standard dictionary. The first one is how to define a push-forward of arithmetic cycles in our situation. If fQ : XQ → YQ is smooth, then, according to Gillet-Soulé’s arithmetic intersection theory [9], we can get the push-forward f∗ : ĈH p+1 (X) → ĈHp(Y ). We would not like to restrict ourselves to the case where fQ is smooth because in the geometric case, the weak positivity of disX/Y (E) gives wonderful applications to analyses of the boundary of the moduli space of stable curves. Thus the usual push-forward for arithmetic cycles is insufficient for our purpose. A difficulty in defining the push-forward arises from a fact: if fC : X(C) → Y (C) Date: 25/February/1998, 3:30PM, (Version 3.0). 1991 Mathematics Subject Classification. 14G40.