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

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.