3. Abelian varieties case [04YG]

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3. Abelian varieties case

We identify our tropical geometric compactification $\overline{A_{g}}^{\rm T}$ ([Od16]) of $A_{g}$ with the adjoint type Satake compactification.

Theorem 3.1.

There are canonical homeomorphisms between the three compactifications

\overline{A_{g}}^{\rm T}\cong\overline{A_{g}}^{\rm Sat,\tau_{\rm ad}}\cong\overline{A_{g}}^{\rm MSBJ},

extending the identity on $A_{g}$ .

The second canonical homeomorphism is a special case of Theorem 2.1 and the first is essentially reduced to matrix computations.

In [OO], we also give a purely moduli-theoritic reexplanation of the structure theory of one parameter degenerations of abelian varieties in [Mum72], [FC90], after the above Theorem 3.1 as follows.

Theorem 3.2.

Take a holomorphic maximally degenerating family of principally polarized abelian varieties $\pi\colon(\mathcal{X},\mathcal{L})\to\Delta$ . Consider the rescaled Gromov-Hausdorff limit $B(\mathcal{X},\mathcal{L})$ of diameter $1$ as in Theorem 1.1 ([Od16]) and its discrete Legendre transform $\check{B}(\mathcal{X},\mathcal{L})$ ([GS11], [KS04]).

Then we can enhance the underlying integral affine structure of $\check{B}(\mathcal{X},\mathcal{L})$ as $K$ -affine structure (in the sense of [KS04, §7.1]) naturally via the data of $\pi$ . Furthermore, such $K$ -affine structure recovers $\pi$ up to an equivalence relation generated by base change (replace $t$ by $t^{a}$ with $a\in\mathbb{Q}_{>0}$ ).