Canonical key formula for projective abelian schemes

TL;DR

This paper refines the canonical key formula for projective abelian schemes, extending it to Arakelov geometry and establishing canonical isomorphisms with metrics under certain conditions.

Contribution

It proves a refined, canonical version of the key formula for projective abelian schemes and extends the discussion to Arakelov geometry with metric considerations.

Findings

Established a canonical isomorphism involving the determinant of pushforward line bundle and the sheaf of differentials.

Extended the key formula to include canonical metrics in the Arakelov geometric setting.

Demonstrated the existence of a canonical isometry under smoothness and separation conditions.

Abstract

In this paper we prove a refined version of the canonical key formula for projective abelian schemes in the sense of Moret-Bailly, 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 $\omega_A$ the determinant of the sheaf of differentials of $\pi$ and by $d$ the rank of the locally free sheaf $\pi_*L$. 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 is compatible with arbitrary base-change; (ii). if the generic fibre of $S$ is separated and smooth, then there exist positive integer $m$, canonical metrics on $L$ and on $\omega_A$ such that there exists an isometry {\rm det}(\pi_*\bar{L})^{\otimes 2m}\cong (e^*\bar{\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 $L$.