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, 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 ${\pi }_{\ast }L$. In this paper, we shall prove the following results: (i). there is an isomorphism {\rm det}(π_*L)^{\otimes 24}\cong (e^*ω_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}(π_*\bar{L})^{\otimes 2m}\cong (e^*\barω_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 $L$.

30 pages

References in corpus (1)

• Generalized holomorphic analytic torsion