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: Theorem 1. Let f : A→B be a non-isotrivial semi-abelian scheme of relative dimension g with projective generic fiber. Then deg(W ) ≤ (g − g0) 2 (2γ − 2+ s), where g0 is the dimension of the fixed part and s is the number of non-projective fibers. Note that the degree is zero in the isotrivial case, so that the inequality still holds except for the obvious exception of γ =0 . The method of proof is an easy extension of Moret-Bailly’s (and Szpiro’s [10], which gives our general result for g = 1), and is likely to be known to experts. The reason it might still be worth writing down in full is because of the recently emerging connection with the ABC conjectures. That is, let Ag,n be the moduli space of principally polarized abelian varieties of dimension g with full level-n structure. For n ≥ 3, we have that Ag,n is of log-general type. In fact, according to Mumford ([8], Proposition 3.4), Ag,n has a toroidal compactification Ag,n � → Ag,n, such that the compactification divisor D has normal crossings and