Semi-stable extensions on arithmetic surfaces

Let S be a smooth projective curve over the complex numbers and X → S a semi-stable projective family of curves. Assume that both S and the generic fiber of X over S have genus at least two. Then the sheaf of absolute differentials ΩX defines a vector bundle on X which is semi-stable in the sense of Mumford-Nakano with respect to the canonical line bundle on X . The Bogomolov inequality c1(Ω 1 X) ≤ 4 c2(Ω 1 X) leads to an upper bound for the self-intersection c1(ωX/S) 2 of the relative dualizing sheaf ωX/S . Assume now that S is the spectrum Spec (OF ) of the ring of integers in a number field F and that X → S is a semi-stable (regular) curve over S, with generic genus at least two. In [7], Parshin asked for a similar upper bound for the arithmetic self-intersection ĉ1(ω̄X/S) 2 of the relative dualizing sheaf of X over S, equipped with its Arakelov metric. He and Moret-Bailly [5] proved that a good upper bound for this real number ĉ1(ω̄X/S) 2 would have beautiful arithmetic consequences (including the abc conjecture).