Duality invariance of Faltings heights, Hodge line bundles and global periods

TL;DR

The paper proves duality invariance for abelian varieties over global fields: dual varieties $A$ and $B$ have equal Faltings heights $h(A)=h(B)$ and canonically isomorphic Hodge line bundles $\omega_A\simeq\omega_B$, with these isomorphisms preserving both the Faltings and BSD metrized structures. It shows that complex and real period data are preserved under duality, aligning the global period with the Birch–Swinnerton-Dyer framework. The proofs combine recent work on height differences via Tate–Shafarevich invariants (GRS24, Suz20a/20b) with base-change conductor dualities (OS23) and a detailed study of the determinants of Lie algebras, extending known results to non-semistable function-field cases and providing canonical, functorial identifications of Hodge data under duality.

Abstract

We prove that an abelian variety and its dual over a global field have the same Faltings height and, more precisely, have isomorphic Hodge line bundles, including their natural metrized bundle structures. More carefully treating real places, we also show that these abelian varieties have the same real and global periods that appear in the Birch-Swinnerton-Dyer conjecture.

Theorems & Definitions (35)

• Theorem 1.1: Theorem \ref{['0014']}
• Theorem 1.2: Theorem \ref{['0004']}
• Theorem 1.3: Theorems \ref{['0012']}, \ref{['0031']}
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4