Ampleness of non-symmetric line bundles on degenerations of abelian varieties

Abstract

Answering a 1985 question of Moret-Bailly, we prove by way of counterexample that if a line bundle $\mathscr{L}$ on a degeneration of abelian varieties $G \to \mathrm{Spec}(R)$ is not symmetric, then $\mathscr{L}$ being ample on $G$ is not necessarily equivalent to $\mathscr{L}$ being ample on the generic fiber of $G$. Our technique of proof is to use the Fourier expansion principle for sections of line bundles on degenerations, due to Faltings--Chai and Lan.