Canonical key formula for projective abelian schemes

Abstract

In this paper we prove a refined version of the canonical key formula for projective abelian schemes in the sense of Moret-Bailly (cf. Asterisque 129, 1985), we also extend this discussion to the context of Arakelov geometry. Precisely, let \({\pi: A \to S}\) be a projective abelian scheme over a locally noetherian scheme S with unit section \({e: S \to A}\) and let L be a symmetric, rigidified, relatively ample line bundle on A. Denote by ωA the determinant of the sheaf of differentials of π and by d the rank of the locally free sheaf π*L. In this paper, we shall prove the following results: (i). there is an isomorphism $${\rm det}(\pi_*L)^{\otimes 24} \cong (e^*\omega_A^\vee)^{\otimes 12d}$$ which is canonical in the sense that it can be chosen to be functorial, namely it is compatible with arbitrary base-change; (ii). if the generic fibre of S is separated and smooth, then there exist a positive integer m and canonical metrics on L and on ωA such that there exists an isometry $${\rm det}(\pi_*\overline{L})^{\otimes 2m} \cong (e^*\overline{\omega}_A^\vee)^{\otimes md}$$ which is canonical in the sense of (i). Here the constant m only depends on g, d and is independent of L.