Subsection [04XJ]

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(3.7) If $A$ has purely toric reduction, then we can interpret $\rho_{A}$ as a non-archimedean SYZ fibration by means of the theory of Mumford models [Mu72] and the refinements of Mumford’s construction given in [Kü98]. We say that a model $\mathscr{P}$ of $A$ is a Künnemann-Mumford model if it is a regular model that arises through the construction in the proof of [Kü98, 3.5].

Proposition 3.8.

Let $A$ be an abelian $K$ -variety of dimension $n$ . Then the essential skeleton $\mathrm{Sk}(A)$ of $A$ coincides with Berkovich’s canonical skeleton $\Delta(A)$ . If $A$ has semi-abelian reduction and $\mathscr{P}$ is a Künnemann-Mumford model of $A$ over $R$ , then $\mathscr{P}$ is a good minimal dlt-model that satisfies assumption (2). If $A$ has purely toric reduction, then the non-archimedean SYZ fibration $\rho_{\mathscr{P}}$ coincides with Berkovich’s canonical retraction $\rho_{A}$ . In particular, $\rho_{\mathscr{P}}$ is an $n$ -dimensional affinoid torus fibration.

Proof.

The equality $\Delta(A)=\mathrm{Sk}(A)$ is proven in [HN17, 4.3.2]. Let $\mathscr{P}$ be a Künnemann-Mumford model for $A$ over $R$ . Then, by definition, $\mathscr{P}$ is an snc-model, and thus certainly good and dlt. It is shown in [HN17, 5.1.7] that $\mathscr{P}$ is minimal.

Let $\widetilde{\mathscr{P}}$ be a regular relatively complete model of $T$ as in [Kü98, 2.11] such that the formal $t$ -adic completion of $\mathscr{P}$ arises as a quotient of the formal $t$ -adic completion of $\widetilde{\mathscr{P}}$ under an action of the period lattice. Then, by construction, $\widetilde{\mathscr{P}}$ is a torus embedding of $T$ over $R$ , and we have a commutative diagram

Thus in order to prove that $\rho_{\mathscr{P}}=\rho_{A}$ , it suffices to observe that $\rho_{\widetilde{\mathscr{P}}}=\rho_{T}$ by Example 3.5. ∎

Remark 3.9.

A refinement of the proof shows that the equality $\rho_{A}=\rho_{\mathscr{P}}$ remains valid if we only assume that $A$ has semi-abelian reduction; then the non-archimedean uniformization of $A$ takes the form $\pi:E^{\mathrm{an}}\to A^{\mathrm{an}}$ , where $E$ is an extension of an abelian $K$ -variety $B$ with good reduction by a split $K$ -torus $T$ . The dimension of $T$ is precisely the toric rank of $\mathscr{A}^{o}_{k}$ , the identity component of the special fiber of the Néron model of $A$ . The Künnemann-Mumford construction produces a relatively complete model $\widetilde{\mathscr{P}}$ of $E$ that is a Zariski-locally trivial fibration in torus embeddings over the Néron model of $B$ . Since we do not need this generalization in this paper, we omit the details.