Relative compactification of semiabelian N\'eron models, II

Abstract

Let $R$ be a complete discrete valuation ring, $k(\eta)$ its fraction field, $S={\rm Spec} R$, $(G_{\eta},\mathcal{L}_{\eta})$ a polarized abelian variety over $k(\eta)$ with $\mathcal{L}_{\eta}$ symmetric ample cubical and $\mathcal{G}$ the N\'eron model of $G_{\eta}$ over $S$. Suppose that $\mathcal{G}$ is semiabelian over $S$. Then there exists a {\it unique} relative compactification $(P,\mathcal{N})$ of $\mathcal{G}$ such that ($\alpha$) $P$ is Cohen-Macaulay with codim$_P(P\setminus\mathcal{G})=2$ and ($\beta$) $\mathcal{N}$ is ample invertible with $\mathcal{N}_{|\mathcal{G}}$ cubical and $\mathcal{N}_{\eta} = \mathcal{L}^{\otimes n}_{\eta}$ for some positive integer $n$. The totally degenerate case has been studied in \cite{MN24}. We discuss here first the partially degenerate case and then the case where $R$ is a Dedekind domain.