ABC inequalities for some moduli spaces of log-general type

Let B be a smooth projective curve of genus γ over the complex numbers and let f : A→B be a non-isotrivial semi-abelian scheme over B with projective generic fiber of relative dimension g. Let U ⊂ B be the locus above which the fibers are projective, and let S = B−U (a finite set). Thus f : AU→U is abelian, and f : A→B is the connected component of its Neron model. Denote by g0 the dimension of the fixed part of A and s = |S|. We will adopt the convention of using the same notation for the map f and several of its restrictions, unless an explicit danger of confusion forces us to do otherwise. Let e : B→A be the identity section, and let W := e∗ΩA/B . Various authors have dealt with upper and lower bounds for the degree of W . Faltings [5], for example, shows that deg(W ) ≤ g(3γ + s + 1) while Moret-Bailly [7] shows that deg(W ) ≤ (g−g0)(γ−1) in the case where A/B is smooth. Arakelov [1] had earlier given the bound (g − g0)(γ − 1 + s/2) when A is the connected component of the Jacobian of a family of stable curves. In this paper, we improve a bit on Faltings, in the general case: