Logarithmic good reduction of abelian varieties

Abstract

Let K be a field which is complete for a discrete valuation. We prove a logarithmic version of the Néron–Ogg–Shafarevich criterion: if A is an abelian variety over K which is cohomologically tame, then A has good reduction in the logarithmic setting, i.e. there exists a projective, log smooth model of A over $\mathcal {O}_K$. This implies in particular the existence of a projective, regular model of A, generalizing a result of Künnemann. The proof combines a deep theorem of Gabber with the theory of degenerations of abelian varieties developed by Mumford, Faltings–Chai et al.

Appendix: a fibrewise criterion for log smoothness

We used the following lemma, which is more general than needed. It may however be of independent interest (we could not locate such a result in the literature).

Lemma  Let $f :(X,\mathcal {M}_X)\rightarrow (T,\mathcal {M}_T)$ be an integral morphism of fine log schemes with log smooth fibres, such that the morphism of underlying schemes $X \rightarrow T$ is locally of finite presentation and flat. Then f is log smooth.

To prove this, we use Olsson’s theory of stacks of log structures [14].

Proof

By [14, 4.6.(ii)], it suffices to show that the induced morphism

$$\begin{aligned} \mathcal {L}og(f):\mathcal {L}og_X\rightarrow \mathcal {L}og_T \end{aligned}$$

(3)

of algebraic stacks is formally smooth. By our assumptions, it is locally of finite presentation; hence it suffices to show that (3) is flat and that its geometric fibres are smooth.

We will prove the last statement first. Given a scheme Y, denote by $\underline{Y}$ the associated stack. Let $\underline{t} \rightarrow \mathcal {L}og_T$ be a geometric point. This yields a morphism of fine log schemes $(t,\mathcal {M}_t) \rightarrow (T,\mathcal {M}_T)$, which need not be strict: one obtains $(t,\mathcal {M}_t)$ as the image of $\mathrm {Id} \in \underline{t}(t)$ in $\mathcal {L}og_T$. Now $(X_{t}, \mathcal {M}_{X_{t}})$ is given by the cartesian diagram

(in the category of fine log schemes) and the underlying scheme $X_t$ is the fibre product in the category of schemes, since the morphism $(X,\mathcal {M}_X)\rightarrow (T,\mathcal {M}_T)$ is integral. By our assumptions on the fibres, the morphism $(X_t,\mathcal {M}_{X_t})\rightarrow (t,\mathcal {M}_t)$ is log smooth. In particular, [14, 4.6.(ii)] implies that $\mathcal {L}og_{X_t}\rightarrow \mathcal {L}og_t$ is smooth. By [14, 3.20], we have an isomorphism

$$\begin{aligned} \mathcal {L}og_{X_t}\cong \mathcal {L}og_X\times _{\mathcal {L}og_T}\mathcal {L}og_t. \end{aligned}$$

Base change along the open immersion $\underline{t}\rightarrow \mathcal {L}og_t$ shows that the induced morphism

$$\begin{aligned} \mathcal {L}og_X\times _{\mathcal {L}og_T}\underline{t} \rightarrow \underline{t} \end{aligned}$$

is indeed smooth. Hence it remains to show that (3) is flat, or that f is log flat. Since log smoothness implies log flatness, this follows from [2, Theorem 2.6.3]. $\square$