TY - JOUR
AU1 - Odaka,, Yuji
AB - Abstract We compactify the classical moduli variety Ag of principally polarized abelian varieties of complex dimension g, by attaching the moduli of flat tori of real dimensions at most g in an explicit manner. Equivalently, we explicitly determine the Gromov–Hausdorff limits of principally polarized abelian varieties. This work is analogous to [50], where we compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov–Hausdorff limits along holomorphic families of abelian varieties and show that these form special nontrivial subsets of the whole boundary. We also do the same for algebraic curves case and observe a crucial difference with the case of abelian varieties. 1 Introduction This paper is a companion paper to (or sequel of) [50], in which we gave a couple of compactifications of the moduli of hyperbolic projective curves Mg and analyzed them. In this paper, in turn, we work on the moduli space Ag of g-dimensional principally polarized complex abelian varieties. What we first prove in our Section 2 can be roughly summarized as follows. Theorem 1.1. (cf., 2.1, 2.3, 2.5). The moduli space Ag of principally polarized abelian varieties over |$\mathbb {C}$| with the complex analytic topology admits a compactification |$\bar {A}_{g}^{T}$| that attach (as its boundary) the moduli spaces of all real flat tori of dimension 1 up to g, the half of the original abelian varieties’ real dimension. We interpret the real flat tori that appear here, as tropical abelian varieties (cf., e.g., [11, 38]). For this reason, we would like to call the compactification |$\bar {A}_{g}^{T}$| the tropical geometric compactification of Ag. However, in this particular case of abelian varieties, in reality the compactification is nothing but the “Gromov–Hausdorff compactication” that is attaching the moduli space of all possible Gromov–Hausdorff limits as metric spaces, to the original moduli space (Ag), as boundary. The abstract existence of such Gromov–Hausdorff compactification of Ag, without explicitly knowing what are the actual limits and the structure of the boundary, is a direct consequence of the well-known Gromov’s precompactness theorem [24]. Therefore, our point is to study the very explicit structures of the compactification by in particular identifying the Gromov–Hausdorff limits as specific flat tori and also show relations with other fields. Remark 1.2. The reason why we would like to allow two possible names for the identical compactification above, tropical geometric compactifications and Gromov–Hausdorff compactifications, is as follows. Basically the author sees this coincidence as a special phenomenon that happens only for abelian varieties, etc. As in the case of curves [50], the boundaries of the Gromov–Hausdorff compactifications parametrize only metric space. On the other hand, what we would like to propose under the name of tropical geometric compactifications (of each moduli space of collapsing Kähler–Einstein-polarized varieties) should in general parametrize a priori more information, such as affine structures, to regard the collapses as “polarized tropical varieties” rather than simply metric spaces. In that way, the author believes the compactifications should become more tractable and comes with richer structures. At the moment of writing this, we only have case-by-case “working definitions” of such compactifications for classical cases as Mg [50], Ag, the moduli spaces of K3 surfaces, and hyperkähler varieties [51]. See Remark 2.6 for more explanation for our particular case. More precisely, the point of Section 2 of this paper is, by using the Siegel reduction and so on, to explicitly determine the Gromov–Hausdorff limits as well as the structure of the compatification. Then, we continue to make some more basic analysis of the compactification including the relations of cohomologies and homologies. Then, in Section 3 in turn, we determine the holomorphic limits and compare. That is, we consider Gromov–Hausdorff limits of an arbitrarily given punctured holomorphic family of either principally polarized abelian varieties or of canonically polarized curves of the form |$(\mathcal {X},\mathcal {L})\to \Delta ^{\ast }$|⁠, where Δ* denotes a punctured unit disk |$\{t\in \mathbb {C}\mid 0<|t|<1\}$|⁠. The following statements, in which we summarize some results of our Section 3, show that the Gromov–Hausdorff limit does not depend on the sequence we take once we fix the family. For (i), we only prove under an assumption (triviality of the Raynaud extension) in this paper, and the full extension to the following generality with its proof will be presented in our forthcoming joint paper [51] with Y. Oshima. Theorem 1.3. (i) (cf., 3.1, also [51]) Let us fix a punctured meromorphic family of g-dimensional principally polarized abelian varieties |$(\mathcal {X},\mathcal {L})\to \Delta ^{\ast }\ni t$| and consider a sequence t(i) (where i = 1, 2, ⋯ ) ∈Δ* converging to 0. Then the Gromov–Hausdorff limit of |$\mathcal {X}_{t(i)}$| with rescaled flat Kähler metric of diameter 1 at 0 ∈ Δ exists, which does not depend on the choice of {t(i)}i. Furthermore, such limits form the union of Ag and a dense subset (consists of “rational points”) inside the whole boundary |$\partial \bar {A}_{g}^{T}$|⁠. (ii) (cf., 3.2.1, 3.13) Let us fix a punctured algebraic family of smooth projective curves of genus g ≥ 2 |$\mathcal {X}\to \Delta ^{\ast }\ni t$| and consider a sequence t(i)(i = 1, 2, ⋯ ) ∈ Δ* converging to 0. Then, the Gromov–Hausdorff limit of |$\mathcal {X}_{t(i)}$| with rescaled Poincaré metric of diameter 1 exists, which does not depend on the choice of {t(i)}i. Furthermore, such limits form the union of Mg and a finite subset inside the whole real 3g−4 dimensional boundary |$\partial \bar {M}_{g}^{T}$|⁠. Finally, our appendix discusses the Morgan–Shalen compactification [41], which have been recently revisited and extended by Favre [22] and Boucksom and Jonsson [10]. We introduce a technically subtle extension and prove basic properties, to apply to our compactifications in later works (in e.g., 3.7, [51]). As our series of papers heavily depends on the basic theory of Gromov–Hausdorff convergence, we refer to a textbook [12] if needed. 2 Compactifying Ag 2.1 Gromov–Hausdorff collapse of abelian varieties In this section, to each g-dimensional principally polarized abelian varieties (V, L), we associate a rescaled Kähler–Einstein metrics whose diameters are 1. That is, we consider the flat Kähler metric gKE whose Kähler class is c1(L) and consider the induced distance on V, which we denote by dKE(V ) in this paper. Then we rescale to |$\frac {d_{\textrm {KE}}}{\textrm {diam}(d_{\textrm {KE}})}$| of diameter 1, which will be the metric in concern. diam(−) means the diameter. Note that in this case, precompactness of the corresponding moduli space Ag with respect to the associated Gromov–Hausdroff distance follows from the famous Gromov’s precompactness theorem [24] (while it also directly follows from our arguments in this section). We sometimes omit the principal polarization and simply write principally polarized abelian varieties as V or Vi as far as it should not cause any confusion. For the basics of the Gromov–Hausdorff convergence in metric geometry, we refer to, for example, the textbook [12]. Now, we proceed to classification of all the possible Gromov–Hausdorff limits of them. The author suspects it has been naturally expected by experts and at least partially known that those collapse should be (real) flat tori but unfortunately he could not find precise study nor results in literatures. So we present here a precise statement as well as its proof and also give explicit determinations of the limits. In particular, our arguments show that the flat tori, which can appear as such Gromov–Hausdorff limits, have its real dimension at most g (which is the half of the real dimension of the original complex abelian varieties) and is characterized only by that condition. For simplicity and better presentation of ideas, let us first restrict our attention to maximally degenerating case and establish the general case later. Theorem 2.1. Consider an arbitrary sequence of g-dimensional principally polarized complex abelian varieties {Vi}i=1, 2, 3, ⋯, which is converging to the cusp A0 of the boundary |$\partial \bar {A_{g}}$|さBB of the Satake–Baily–Borel compactification |$\bar {A_{g}}$|さBB. The character “さ” is Hiragana-type character, which we pronouce “SA”, the first syllable of Satake and the idea of using this character, is after Namikawa’s book [45], which used Katakana “サ” instead but we Japanese rarely use katakana for writing Japanese name. The corresponding Kanji character 佐 is more commonly used. We denote the flat (Kähler) metrics with respect to the polarization dKE(Vi) and their diameters diam(dKE(Vi)). Then, after passing to an appropriate subsequence, we have a Gromov–Hausdorff limit of |$\left\{\left(V_{i}, \frac {d_{\textrm {KE}}(V_{i})}{\textrm {diam}(d_{\textrm {KE}}(V_{i}))}\right)\right\}_{i}$|⁠, which is (g − r)-dimensional (flat) tori of diameter 1 with some (0 ≤)r(< g). Conversely, any such flat (g − r)-dimensional torus of diameter 1 with 0 ≤ r < g can appear as a possible Gromov–Hausdorff limit of such sequence of g-dimensional principally polarized abelian varieties with fixed diameter 1. Proof. Let us first set up our notations (mainly after [15]) on the Siegel upper half space and its compactification theory due to I. Satake [52]. In our proof, we make essential use of the Siegel reduction theory. For a point |$Z=X+\sqrt {-1}Y$| of the Siegel upper half space |$\mathfrak {H}_{g}$|⁠, we denote the Jacobi decomposition of Y as Y = tBDB, where $$\begin{align*} B=&\,\left(\begin{array}{ccccc} 1 & b_{1,2} & b_{1,3} & \cdots & b_{1,g} \\ & 1 & b_{2,3} & \cdots & b_{2,g} \\ & & 1 & \cdots & \vdots \\ &\textrm{0} & & \ddots & \vdots\\ &&&&1 \\ \end{array}\right),\\ D= \textrm{diag}(d_{1},\cdots,d_{g})=&\, \left(\begin{array}{ccccc} d_{1} &&&&\\ & d_{2} & & \textrm{0}&\\ && d_{3} & &\\ &\textrm{0}&& \ddots &\\ &&&&d_{g}\\ \end{array}\right). \end{align*}$$ Equivalently, writing |$Y={}^{t}\sqrt {Y}\sqrt {Y}$| with a matrix |$\sqrt {Y}\in \textrm {GL}(g,\mathbb {R})$|⁠, $$\begin{align*} \sqrt{Y}=\textrm{diag}\left(\sqrt{d_{1}},\cdots,\sqrt{d_{g}}\right)B\end{align*}$$ is the corresponding Iwasawa decomposition. Using the above notation, recall that the Siegel subset |$\mathfrak {F}_{g}(u)$| of the Siegel upper half plane |$\mathfrak {H}_{g}$| is defined as $$\begin{align} \left\{X+\sqrt -1 Y \in \mathfrak{H}_{g} \mid |x_{ij}|<u, |1-b_{i,j}|<u, 1<ud_{1}, d_{i}<ud_{i+1}\ \textrm{for all}\ i,j\right\}. \end{align}$$ (1) It is known to satisfy $$\begin{align} \textrm{Sp}_{2g}(\mathbb{Z})\cdot\mathfrak{F}_{g}(u)=\mathfrak{H}_{g} \end{align}$$ (2) for u ≫ 0. Let us set $$\begin{align*} \mathfrak{H}_{g}^{\ast}:=\mathfrak{H}_{g}\sqcup\mathfrak{H}_{g-1}\sqcup\mathfrak{H}_{g-2}\sqcup\cdots \sqcup \mathfrak{H}_{0}.\end{align*}$$ Then the Satake–Baily–Borel compactification |$\bar {A_{g}}$|さBB [52] can be defined as |$\mathfrak {H}_{g}^{\ast }/\sim $| with some equivalent relation ∼ extending |$\textrm {Sp}_{2g}(\mathbb {Z})$|-action on |$\mathfrak {H}_{g}$|⁠. Thanks to above (2), we can suppose that, fixing sufficiently large u0 ≫ 1, we have |$\bar {A_{g}}$|さBB|$ =\mathfrak {F}_{g}^{*}(u_{0})/\sim $| with the same equivalent relation, where $$\begin{align*} \mathfrak{F}_{g}^{\ast}(u_{0}):= \mathfrak{F}_{g}(u_{0})\sqcup \mathfrak{F}_{g-1}(u_{0})\sqcup \cdots \mathfrak{F}_{0}(u_{0}) \subset \mathfrak{H}_{g}^{\ast}. \end{align*}$$ We refer to [15] for the details of above discussions. Using the above reduction, we can assume that the whole sequence {Vi}i=1, 2, ⋯ of principally polarize abelian varieties are parametrized by a sequence |$Z_{i}=X_{i}+\sqrt {-1}Y_{i}$| in |$\mathfrak {F}_{g}(u_{0})$| for the fixed u0, which is sufficiently large. Thanks to such boundedness, appropriately passing to a subsequence, we can and do assume Xi and Bi converges. Our principally polarized abelian variety that corresponds to |$Z=X+\sqrt {-1}Y\in \mathfrak {H}_{g}$| is $$\begin{align*} \mathbb{C}^{g}\left/ \left(\begin{array}{cc} 1 & X \\ & Y \\ \end{array}\right)\right.\mathbb{Z}^{2g},\end{align*}$$ which is isometric to |$\mathbb {R}^{2g}/\mathbb {Z}^{2g}$| with metric matrix $$\begin{align} \left(\begin{array}{cc} 1 & \\ X & Y \\ \end{array}\right) \left(\begin{array}{cc} Y^{-1} &\\ & Y^{-1} \\ \end{array}\right)\left(\begin{array}{cc} 1 & X \\ & Y \\ \end{array}\right)= \left(\begin{array}{cc} Y^{-1} & Y^{-1}X \\ XY^{-1}& XY^{-1}X+Y\\ \end{array}\right). \end{align}$$ (3) If we denote the corresponding standard basis of the lattice |$\left ( 1 \quad {{X} \atop {Y}}\right ) \mathbb {Z}^{2g}$|as e1, ⋯ , e2g and the Kähler–Einstein metric as gKE as before, then what we meant by metric matrix is defined as {gKE(ei, ej)}1≤i, j≤2g. In particular, if X = (0) (zero matrix), then the metric matrix of our torus |$\mathbb {R}^{2g}/\mathbb {Z}^{2g}$| is $$\begin{align*} \left(\begin{array}{cc} Y^{-1} & \\ & Y \\ \end{array}\right). \end{align*}$$ Thus what we would like to classify are possible Gromov–Hausdorff limits of $$\begin{align*} \frac{1}{\textrm{diam}(V_{i})}\cdot \left(\begin{array}{cc} Y_{i}^{-1} & Y_{i}^{-1}X_{i} \\ X_{i}Y_{i}^{-1}& X_{i}Y_{i}^{-1}X_{i}+Y_{i}\\ \end{array}\right). \end{align*}$$ We set our notation as follows. For our points |$Z_{i}=X_{i}+\sqrt -1 Y_{i}$| in the Siegel set |$\mathcal {F}_{g}(u_{0})$|⁠, which give our principally polarized abelian varieties Vi (i = 1, 2, ⋯), we do the Jacobi decomposition of |$\sqrt Y_{i}$| and denote the corresponding djs as dj(Vi). Replacing diam(Vi) by dg(Vi), let us first classify possible limits of $$\begin{align*} \frac{1}{d_{g}(V_{i})}\cdot \left(\begin{array}{cc} Y_{i}^{-1}& Y_{i}^{-1}X_{i}\\ X_{i}Y_{i}^{-1}& X_{i}Y_{i}^{-1}X_{i}+Y_{i}\\ \end{array}\right) \end{align*}$$ instead. Our assumption that it converges to the cusp |$A_{0}\in \partial \bar {A_{g}}$|さBB (“maximally degenerating”) is equivalent to that |$d_{1}(V_{i})\rightarrow +\infty $| when |$i\rightarrow +\infty $| from the definition of Satake topology (cf., [15, 52]). Now, let us set $$\begin{align*} r:=\min\left\{(1\le)j(\le g)\mid \lim\inf_{i\rightarrow +\infty} \frac{d_{j}(V_{i})}{d_{g}(V_{i})}>0\right\}-1. \end{align*}$$ Then, after appropriately passing to a subsequence again, we can assume that |$\frac {d_{r}(V_{i})}{d_{g}(V_{i})}\rightarrow +0$| so that |$\frac {d_{j}(V_{i})}{d_{g}(V_{i})}\rightarrow +0$| for all j ≤ r. By once more replacing {Vi}i by a subsequence if necessary, we can assume without loss of generality that for each 1 ≤ j ≤ g − r the sequence |$\left\{\frac {d_{r+j}(V_{i})}{d_{g}(V_{i})}\right\}_{i}$| converges. We denote convergence values as ar+j for each j. We prove that then |$\left\{(V_{i},\frac {d_{\textrm {KE}}(V_{i})}{d_{g}(V_{i})})\right\}_{i}$| converges to a (g − r)-dimensional torus as |$i\to +\infty $|⁠. As |$d_{g}(V_{i})\rightarrow +\infty $|⁠, it follows that |$Y_{i}^{-1}/d_{g}(V_{i})\rightarrow +0$|⁠. On the other hand, thanks to our preceded set of processes of replacing {Vi}i by its subsequence, the following holds when |$i\rightarrow +\infty $|⁠. Please note that the down arrow between the above big matrices “↓” means convergence as g × g real matrices, when |$i\to \infty $|⁠. From the above convergence of the matrices, it follows straightforward that |$\mathbb {R}^{2g}/\mathbb {Z}^{2g}$| whose metric matrix is $$\begin{align*} \frac{1}{d_{g}(V_{i})} \left(\begin{array}{cc} Y_{i}^{-1}& Y_{i}^{-1}X_{i}\\ X_{i}Y_{i}^{-1}& X_{i}Y_{i}^{-1}X_{i}+Y_{i}\\ \end{array}\right) \end{align*}$$ converges to a (g − r)-dimensional torus in the Gromov–Hausdorff sense, when |$i\to +\infty $|⁠. From this result, we particularly deduce the following. Claim 2.2. In the above setting, we have $$\begin{align*} d_{g}(V_{i})\sim d_{\textrm{KE}}(V_{i}),\end{align*}$$ that is, the ratio of the left-hand side and the right-hand side is bounded on both sides (by some positive constants) when |$i \to +\infty $|⁠. Going back to proof of Theorem 2.1, now we would like to prove the other direction that is to show that every (g − r)-dimensional flat torus with 0 ≤ r ≤ g of diameter 1 can indeed appear as the above type Gromov–Hausdorff limit. Indeed, we can construct such a sequence in the following explicit manner, for instance. Fix |$(a_{r+1},\cdots ,a_{g})\in \mathbb {R}_{>0}^{g-r}$|⁠. Then set a sequence of g × g diagonal real matrices {Di}i=1, 2, ⋯ as $$\begin{align*} D_{i}:= \textrm{diag}(d_{1,i},\cdots,d_{g,i}):= \left(\begin{array}{ccccc} d_{1,i}&& & &\\ & d_{2,i} & & 0 &\\ & & d_{3,i} &&\\ &0&& \ddots & \\ & & & &d_{g,i}\\ \end{array}\right),\end{align*}$$ where $$\begin{align*} d_{j,i}:= 1 \end{align*}$$ for j ≤ r and $$\begin{align*} d_{j,i}:= (i+1)^{r}a_{j} \end{align*}$$ for j ≥ r + 1. Recall the notation at the beginning of our proof of Theorem 2.1. Let us fix “X-part and B-part”, that is, set |$Z_{i}=X_{i}+\sqrt {-1}Y_{i}$| with constant Xi = X and Bi = B, where Yi =tBiDiBi is the Jacobi decomposition for i = 1, 2, ⋯ and denote the corresponding principally polarized abelian variety as Vi with the associated Kähler–Einstein metric gKE(Vi). Then the Gromov–Hausdorff limit of |$\left (V_{i},\frac {g_{KE}(V_{i})}{d_{g}(V_{i})}\right )$| for |$i\to \infty $| is a (g − r)-dimensional tori whose metric matrix is Letting B run over all upper triangular real g × g matrices, we get all g × g matrices of the form with a positive definite (g − r) × (g − r) symmetric matrix P, as above limits. In such situation, the Gromov–Hausdorff limit of |$\left(V_{i},\frac {g_{KE}(V_{i})}{d_{g}(V_{i})}\right)$| for |$i\to \infty $| is a (g − r)-dimensional flat real torus whose metric matrix is P. Thus we complete the proof. Please note that r < g can really happen while the conjectures [35, Conjecture 1, p. 19] and [25, Conjecture 5.4], motivated by the Strominger–Yau–Zaslow mirror symmetry on Calabi–Yau varieties, expect the collapse only to just half dimensional affine manifolds (with singularities), that is, i = g case. This difference occurs naturally, without contradiction, since we take an arbitrary sequence rather than dealing with proper algebraic family with maximal monodromy as op.cit do. For a general sequence in Ag, we prove the following. Theorem 2.3. We use the same notation as Theorem 2.1. Suppose a sequence of g-dimensional principally polarized complex abelian varieties {Vi}i=1, 2, ⋯ converges to a point of |$A_{c} \subset \partial \bar {A_{g}}$|さBB with 0 ≤ c < g in the Satake–Baily–Borel compactification |$\bar {A_{g}}$|さBB. Then, after passing to a subsequence, |$\left (V_{i}, \frac {d_{\textrm {KE}}(V_{i})}{\textrm {diam}(d_{\textrm {KE}}(V_{i}))} \right )$| converges to a (g − r)-dimensional (flat) tori of diameter 1 with some (c ≤)r(< g), in the Gromov–Hausdorff sense. Conversely, any such flat (g − r)-dimensional torus of diameter 1 with c ≤ i ≤ g can appear as a possible Gromov–Hausdorff limit of such sequence of g-dimensional principally polarized complex abelian varieties with diameter 1 rescaled Kähler–Einstein (flat) metrics. Before going to the proof, let us analyze what the above particularly means. Note that for each fixed c the set of possible limits described above is included in the corresponding limit set of the maximal degeneration case 2.1. Morally speaking, this can be seen as a special case of more general phenomenon that “degeneration/deformation” order get reversed once we pass from algebro-geometric setting to its tropical analog. Indeed, similar phenomenon happened in curve case [50]. Another very simple fact, which is partially related to above, reflecting such general phenomenon is the following. It roughly states that Gromov–Hausdorff limit of degenerating spaces sees just “degenerating part” and “ignores non-degenerating part”. Proposition 2.4. Suppose a sequence of compact metric spaces |$\{X^{(i)}\}_{i\in \mathbb {Z}_{>0}}$| decomposes as $$\begin{align*} X_{1}^{(i)}\times \cdots \times X_{m}^{(i)}\end{align*}$$ as metric spaces with p-product metric for some p > 0. If the last component |$X_{m}^{(i)}$| is “responsible of degeneration” in the sense that (i) |$\textrm {diam}\left (X_{m}^{(i)}\right )\rightarrow +\infty $| and (ii) |$\textrm {diam}\left (X_{j}^{(i)}\right )\le $| constant for all j≠m, then the Gromov–Hausdorff limit “only sees |$X_{m}^{(i)}$|” in the sense that $$\begin{align*} \lim_{i\rightarrow +\infty}\left(X^{(i)}\left/\textrm{diam}\left(X^{(i)}\right)\right)= \displaystyle \lim_{i\rightarrow +\infty}\left(X_{m}^{(i)}\right/\textrm{diam}\left(X_{m}^{(i)}\right)\right). \end{align*}$$ Here the above |$\lim _{i\rightarrow +\infty }$| means the Gromov–Hausdorff limits and |$(X^{(i)}/\textrm {diam}\left (X^{(i)}\right ))$| (resp., |$\left (X_{m}^{(i)}/\textrm {diam}\left (X_{m}^{(i)}\right )\right )$| means the topological space X(i)|$\left (\textrm {resp.}, X_{m}^{(i)}\right )$| with the rescaled metric of the original metric, with diameter 1. A trivial remark is that the statement of the above proposition is just equivalent to m = 2 case but we stated as above just to get a better intuition for various applications. Proof of Proposition 2.4. The whole point is simply that there is a constant c that satisfies the inequality $$\begin{align*} \textrm{diam}\left(X_{m}^{(i)}\right)\le\textrm{diam}\left(X^{(i)}\right)\le\textrm{diam}\left(X_{m}^{(i)}\right)+c \end{align*}$$ for all i. The assertion easily follows from the above. Thus indeed if a punctured family of abelian varieties with semi-abelian reduction with torus rank (g − r) of the central fiber, it follows that the torus part determines the Gromov–Hausdorff limit (with fixed diameters). Theorem 2.3 reflects such a fact. Let us now turn to the proof of Theorem 2.3 that is the classification of our Gromov–Hausdorff limits of principally polarized abelian varieties. Proof of Theorem 2.3. As the proof is a fairly simple extension of the proof of maximal degeneration case (2.1), without bringing essentially new ideas, here we only sketch the proof, focusing on the differences. As in (2.1), thanks to the Siegel reduction theory, we can and do fix sufficiently large u0 ≫ 0 so that our sequence can be parametrized by a sequence $$\begin{align*} \left\{Z_{i}=X_{i}+\sqrt{-1}Y_{i}\right\}_{i=1,2,\cdots}\end{align*}$$ in the Siegel set |$\mathfrak {F}_{g}(u_{0})$| (cf., the definition (1) in the proof of Theorem 2.1). Again in the same manner, we can and do appropriately take a subsequence so that the following conditions hold. (i) Xi converges when |$i\rightarrow +\infty $|⁠, (ii) the upper triangle matrix part B(Vi) converges when |$i\rightarrow +\infty $|⁠, (iii) dj(Vi) for any (1 ≤)j(≤)c converges when |$i\rightarrow +\infty $|⁠, (iv) |$d_{c+j}(V_{i})\rightarrow +\infty $| when |$i\rightarrow +\infty $| for any (1 ≤)j(≤ (g − c)). Here, the notations are same as the proof of (2.1). Let us set again $$\begin{align*} r:=\max\left\{(1\le)j(\le g)\mid \lim\inf_{i\rightarrow +\infty} \frac{d_{j}(V_{i})}{d_{g}(V_{i})}=0\right\}. \end{align*}$$ Then in our general case, we have c ≤ r ≤ g from the definition of the Satake topology [52]. The rest of the proof that Vi converges to a (g − r)-dimensional flat torus with diameter 1 is completely the same. Conversely, for a given r ≥ c, let us prove that any (g − r)-dimensional torus T with diameter 1 can appear as the above limit. From our (2.1), we know there is a sequence of principally polarized abelian varieties (Wi, Mi)i=1, 2, ⋯ of complex dimension g − c, which converges to T in the Gromov–Hausdorff sense. Then, we take arbitrary c-dimensional principally polarized abelian variety W ′ and set Vi := W ′× Wi for each i = 1, 2, ⋯, which admit natural principal polarizations from the construction. Then {Vi}i=1, 2, ⋯ with rescaled Kähler–Einstein metric |$\frac {d_{\textrm {KE}}(V_{i})}{\textrm {diam}(d_{\textrm {KE}}(V_{i}))}$| converge to T in the Gromov–Hausdorff sense, as the simple combination of Proposition 2.4 and Theorem 2.1 show. 2.2 Construction of |$\bar {A}_{g}^{T}$| and comparison with other tropical moduli space Similarly as in the case of curves, we rigorously define our tropical geometric compactification of the moduli space of principally polarized abelian varieties first set-theoretically as $$\begin{align*} \bar{A_{g}}^{T}:=A_{g}\sqcup T_{g},\end{align*}$$ where Tg denotes the set (moduli) of real flat tori with diameters 1 whose dimension is i with 1 ≤ i ≤ g, from now on (T of |$\bar {A}_{g}^{T}$| stands for Tropical while T of Tg stands for Tori.). Then we put a topology on |$\bar {A}_{g}^{T}$| whose open basis can be taken as those of Ag with respect to the complex analytic topology, and metric balls around point [T] in |$\partial \bar {A_{g}}^{T}$| $$\begin{align*} B([T],r):=\left\{[X]\in \bar{A_{g}}^{T} \mid d_{\textrm{GH}}\left([X],[T]\right)<r\right\}, \end{align*}$$ where dGH denotes, as in [50], the Gromov–Hausdorff distance with respect to the rescaled metrics on each flat torus whose diameter is 1. Then we get the following consequence of Theorem 2.3. Corollary 2.5. |$\bar {A}_{g}^{T}$| is a compact Hausdorff space containing Ag as an open dense subset. Remark 2.6. Each k-dimensional real flat torus T (1 ≤ k ≤ g) that is parametrized at the boundary of our |$\bar {A}_{g}^{T}$| carries a canonical integral affine structure by identifying T with |$\mathbb {R}^{k}/\mathbb {Z}^{k}$| and consider the corresponding coordinates (or its equivalent class with respect to the positive scalar rescaling). This is indeed the integral affine structure we should put from the context of the Strominger–Yau–Zaslow mirror symmetry (cf., e.g., [26]). Therefore, such affine structure does not give additional structure that enlarge the Gromov–Hausdorff compactification of Ag. It is the reason why we simply call the above compactification, the tropical geometric compactification. We refer to the discussions in our introduction section. Note that, if we forget complex structures of principally polarized abelian varieties, it gives nontrivial natural morphism from Ag to the moduli space of certain flat tori of real dimension 2g. The latter moduli space has the same dimension as that of Ag (e.g., at the locus, which parametrizes products of g elliptic curves, it gives generally 2g to 1 morphism). Indeed, discreteness of the fibers of the forgetful morphism follows from the fact that, adding marking |$[\pi _{1}(\textrm {complex ab. var of dim} g)\xrightarrow {\,\smash {{{\scriptstyle \simeq }}}\,} \mathbb {Z}^{2g}]$|⁠, which is obviously discrete data, recovers the complex structure of the abelian varieties. It easily follows from the fact that the metric matrix (3) has enough information to recover X and Y. Let us clarify the simple relation with the moduli space |$A_{g}^{tr}$| of tropical abelian varieties constructured in [11]. In their language, the boundary of our tropical geometric compactification |$\bar {A}_{g}^{T}$| is $$\begin{align*} \partial \bar{A}_{g}^{T}\simeq A_{g}^{tr}/\mathbb{R}_{>0}=\left(\Omega^{rt}\setminus \{0\})/(GL(g,\mathbb{Z})\cdot \mathbb{R}_{>0}\right), \end{align*}$$ where |$A_{g}^{tr}$| is the moduli space of g-dimensional tropical (principally polarized) abelian varieties |$\mathbb {R}^{g}/\Lambda $| in the sense of [11], Ωrt (resp., Ω) is the cone of positive semidefinite forms (resp., positive definite forms) on the universal covering |$\mathbb {R}^{g}$| whose null space has a basis inside the rational vector space |$\Lambda \otimes _{\mathbb {Z}} \mathbb {Q}$|⁠, following their notations. Note |$\Omega \subset \Omega ^{rt}\subset \bar {\Omega }$|⁠. Remark 2.7. We make a simple observation on the relation between our Gromov–Hausdorff limits of principally polarized abelian varieties with the dual (intersection) complex (cf., [25, 35]) of algebraic degenerations of them. Such connection is natural, after the well-known conjectures of Kontsevich–Soibelman [35] and Gross–Siebert (cf., [25]) for their approach to the Strominger–Yau–Zaslow conjecture [53]. In their studies, they also predict and partially establish that given a maximal degeneration of general Calabi–Yau manifolds, the dual complex of the special fiber is “close to” the Gromov–Hausdorff limit of Ricci-flat metrics with fixed diameters. Let us think of the relative compactification of Alexeev and Nakamura [3, 5, 42, 43], of a semi-abelian reduction of a generically abelian scheme. Due to [5, (3.17)], [3], and [42, (4.9)], the dual complexes are the duals of the Delaunay triangulations of (g − r)-dimensional tori, which are topologically of course always real torus of (g − r)-dimension (also called “incidence complex” (cf., e.g., [56]), “dual graph”, or “dual intersection complex” (cf., e.g., [25]), etc.). This coincides with our collapsed limits (2.3), except for a slight difference that the tori can get lower dimension as we considered an arbitrary sequence there. We give a closer connection between Alexeev–Nakamura-type degeneration of abelian varieties and our Gromov–Hausdorff limits later in Section 3.1. Similarly, for the case of curves [50], the collapsed limit coincides with the dual graph of the limit stable curves and for higher dimensional semi-log-canonical models, we believe the collapsed limits along horomorphic one parameter degeneration |$\mathcal {X}\twoheadrightarrow \Delta _{t}$| (partially analyzed in [61]) should be at least homeomorphic to the dual complex of lc centers of a log crepant blow up |$\tilde {\mathcal {X}}_{0}$| of |$\mathcal {X}_{0}$| whose normalization |$\tilde {\mathcal {X}}_{0}^{\nu }$| with the conductor divisor cond(ν) is a divisorially log terminal (dlt) pair. However, unfortunately such existence is unknown. Also cf., [34, 5.22]. We also refer to [10] for related recent study. (The author philosophically sees this as a variant of the Yau–Tian–Donaldson correspondence and wishes to come back to this connection at deeper level in future.) 2.3 Finite and infinite joins of Ag Completely similarly as curve case [50], we can naturally construct joins of our tropical geometric compactifications |$\bar {A}_{g}^{T}$|⁠, thanks again to the inductive structure of the boundaries (4). Definition 2.8. The finite join of our tropical geometric compactifications is defined inductively on g as $$\begin{align*} \overline{A_{\leq g}}^{T}:= \overline{A_{\leq(g-1)}}^{T}\cup_{T_{g-1}}\bar{A}_{g}^{T}.\end{align*}$$ The union is obtained via two canonical inclusion maps Tg−1↪Tg and |$T_{g-1}\hookrightarrow \overline {A_{\leq (g-1)}}^{T}$|⁠. We call |$\overline {A_{\leq g}}^{T}$| a finite join of our tropical geometric compactifications. From the definition, we have $$\begin{align*} \cdots\overline{A_{\leq (g-1)}}^{T}\subset \overline{A_{\leq g}}^{T}\cdots.\end{align*}$$ Then we set $$\begin{align*} \bar{A}_{\infty}^{T}:= \lim_{\stackrel{\longrightarrow}{g}} \overline{A_{\leq g}}^{T}=\cup_{g}\overline{A_{\leq g}}^{T}, \end{align*}$$ and call it the infinite join of our tropical geometric compactifications. The boundary of our infinite join |$\bar {A_{\infty }}^{T}$| by which we mean the natural locus |$\cup _{g}(\partial \bar {A}_{g}^{T}=S_{g}^{wt})\subset \bar {A}_{\infty }^{T}$| should be regarded as a tropical version of “|$A_{\infty }$|” introduced and studied recently in [29] a while after the appearance of the first version of this paper (they call it “universal moduli spaces” of abelian varieties). Also note |$\bar {A}_{\infty }^{T}$| is connected and all our tropical geometric compactification |$\bar {A}_{g}^{T}$| is inside this infinite join. 2.4 On the (co)homology groups About the open dense locus Ag, the following has been classically known as a result of A. Borel who proved by studying the vector spaces of |$\textrm {Sp}_{2g}(\mathbb {R})$|-invariant different forms and the group cohomology interpretation that $$\begin{align*} H^{i}(A_{g};\mathbb{Q})=H^{i}\left(\textrm{Sp}_{2g}(\mathbb{Z});\mathbb{Q}\right). \end{align*}$$ Theorem 2.9. ([8]). |$H^{i}(A_{g};\mathbb {Q})=\mathbb {Q}[x_{2},x_{6},x_{10}\ldots ]|_{weight=i}$| for 0 ≤ i < g − 1, where the right-hand side is a polynomial generated by x4a+2 whose weight is 4a + 2. In particular, |$H^{i}(A_{g};\mathbb {Q})=0$| if i is odd, less than g − 1 and the stable cohomology is naturally $$\lim_{\begin{array}{c}\longrightarrow\\ g\end{array}} \ H^{*}(A_{g})=\mathbb{Q}[x_{2},x_{6},x_{10}\ldots ]$$ ⁠. There are also many studies on the homology of symplectic groups such as [16], [39], etc. Using such topological results on Ag, at least partially the study of (co)homologies of the boundary Tg, gives some information on those of |$\bar {A_{g}}^{T}$|⁠. For instance, a simple observation is that dim(Tg) = 3g − 4 combined with the long exact sequence of the Borel–Moore homology groups gives that |$H_{i}(\bar {A}_{g}^{T};\mathbb {Q})=0$| for if i is even and i > g2. Motivated partially from the above discussion, from now on, let us study the boundary Tg. Note that Tg has the following orbifold as an open dense locus $$\begin{align*} \left(\mathbb{R}_{>0}\cdot\textrm{GL}(g,\mathbb{Z})\right)\backslash \Omega,\end{align*}$$ which we will write |${T_{g}^{o}}$|⁠. Then $$\begin{align*} T_{g}=T_{g}^{o}\sqcup T_{g-1}, \end{align*}$$ so that we can partially study the (co)homology of Tg inductively, once we know those of |$T_{g}=(\mathbb {R}_{>0}\cdot \textrm {GL}(g,\mathbb {Z}))\backslash \Omega $|⁠. However, the author does not know well how this cohomology behaves except for the asymptotic behavior of the lower degree due to A. Borel [8], that is $$\begin{align*} H^{i}(T_{g};\mathbb{Q})=H^{i}\left(\textrm{GL}(g;\mathbb{Z});\mathbb{Q}\right)=\mathbb{Q}[x_{3},x_{5},x_{7},\cdots]|_{weight=i}, \end{align*}$$ for |$i\leq \frac {(g-5)}{4}$|⁠. Here, xi has weight i. As in the discussion of the previous paper [50], we have a canonical chain of closed embeddings $$\begin{align} T_{g} \hookrightarrow T_{g+1} \hookrightarrow \cdots, \end{align}$$ (4) which is analogous to the boundary structure of the Satake–Baily–Borel compatification of Ag. We have the following asymptotic triviality of the topologies, analogous to that of curves case [50]. Proposition 2.10. The topological space |$T_{\infty }$| is contractible. |$\textrm {Im}(H_{k}(T_{g};\mathbb {Q})\rightarrow H_{k}(T_{g+1};\mathbb {Q}))=0$| for any k and g. Proof. We imitate the idea of curve case analog in [50] but in this abelian varieties case it is even easier. However, the whole point is still the same, that is to construct an extension |$\psi _{g}\colon CT_{g}\rightarrow T_{\infty }$| of the identity map of Tg where CTg := (Tg × [0, 1])/(Tg ×{1}), which is compatible with lower ψ, that is |$\psi _{g}|_{T_{g-1}}=\psi _{g-1}$|⁠. For ((X, dX), t) ∈ Tg × [0, 1] (dX denotes the flat metric on X), we define $$\begin{align*} \psi_{g}(X,t):=\textrm{rescale of}\ \left(\left(X,(1-t)d_{X}\right)\times S^{1}(t)\right)\ \textrm{with diameter}\ 1. \end{align*}$$ The continuity of the map is obvious. Here, the product means the two-product metric (i.e., simply the square root of the sum of squares of direction-wise distances). It is straightforward to confirm the requirements of the map. Intuitively speaking, the all g-dimensional tori continuously and simultaneously change to once (g + 1)-dimensional tori but later collapse to a circle of circumference 1. On the other hand, we have the following exact sequence from which high nontriviality of the topologies of Tg follows. Proposition 2.11. We have the following two long exact sequences. (i) $$\begin{align*} &\cdots \to H_{k}\left(\textrm{GL}(g;\mathbb{Z});\mathbb{Q}\right)^{\ast}\to H^{k}(T_{g},\mathbb{Q})\to H^{k}(T_{g-1};\mathbb{Q})\to \cdots\\ &\cdots \to H_{k+1}\left(\textrm{GL}(g;\mathbb{Z});\mathbb{Q}\right)^{\ast}\to H^{k+1}(T_{g},\mathbb{Q})\to H^{k+1}(T_{g-1};\mathbb{Q})\to \cdots. \end{align*}$$ (ii) $$\begin{align*}&\cdots \to H_{k}(T_{g-1};\mathbb{Q})\to H_{k}(T_{g},\mathbb{Q})\to H^{k}\left(\textrm{GL}(g;\mathbb{Z});\mathbb{Q}\right)^{\ast}\to \cdots\\ &\cdots \to H_{k-1}(T_{g-1};\mathbb{Q})\to H_{k-1}(T_{g},\mathbb{Q})\to H^{k-1}\left(\textrm{GL}(g;\mathbb{Z});\mathbb{Q}\right)^{\ast}\to \cdots\end{align*}$$ Proof. These are simply the long exact sequences of compactly supported cohomology groups and the Borel–Moore homology groups, respectively, combined with Lefschetz duality for orbifold |$T_{g}\setminus T_{g-1}=\textrm {GL}(g,\mathbb {Z})\backslash \Omega $|⁠. 2.5 Gromov–Hausdorff its with other rescaling There are of course some other ways of rescaling the metrics of abelian varieties that could produce essentially different (pointed) Gromov–Hausdorff its. One of the nontrivial rescaling is (1) via fixing the volume while another is (2) via fixing the injectivity radius. We discuss such two other ways of rescaling but before that, let us illustrate the differences by a simple example. 2.5.1 A simple example Consider again a degenerating sequence of elliptic curves $$\begin{align*} E_{k}:= \mathbb{C}\Big/\left(\mathbb{Z}+\mathbb{Z}k\,\left(a\sqrt{-1}\right)\right) \end{align*}$$ for k = 1, 2, ⋯, while a > 1 fixed. In this case, this is maximally degenerating so that the corresponding “torus rank” is r = 1 = g. The “diameter fixed” Gromov–Hausdorff limit is S1(1/2π) as we observed. Instead, if we fix the injectivity radius, then as the metric is standard metric of |$\mathbb {C}$| we get $$\begin{align*} (\mathbb{R}/\mathbb{Z})\times \left(\sqrt{-1}\mathbb{R}\right)\end{align*}$$ as the pointed Gromov–Hausdorff limit. On the other hand, if we fix the volume of each Ek, then we rescale the metric by multiplying the lengths by |$\frac {1}{\sqrt {ka}}$|⁠. Then the pointed Gromov–Hausdorff limit is the imaginary axis $$\begin{align*} \left(\sqrt{-1}\mathbb{R}\right)\subset \mathbb{C}.\end{align*}$$ In our Gromov–Hausdorff interpretation of the Satake–Baily–Borel compactification |$\mathbb {C}\subset \mathbb {CP}^{1}$| discussed above (2.13), this line of infinite length is corresponding to the cusp |$\{\infty \}$| while the open part |$A_{1}\simeq \mathbb {C}$| parametrizes flat 2-dimensional tori of volume 1. 2.5.2 Fixing the injectivity radius In this subsection, we study pointed Gromov–Hausdorff limits of g-dimensional principally polarized abelian varieties with fixed injectivity radius, which is morally the “minimal” non-collapsing limits. We keep using the previous notation of this section. Recall that for our sequence {Vi}i=1, 2, ⋯ of principally polarized abelian varieties of g-dimension, the corresponding point in the Siegel set is denoted as |$Z_{i}=X_{i}+\sqrt {-1}Y_{i}$| with Yi = tBiDiBi (the Iwasawa decomposition of |$\sqrt {Y_{i}}$|⁠). Similarly as before, after passing to a subsequence, we can and do assume that, for some 0 ≤ r < g, (i) both Xi and Bi converge when i tends to infinity, (ii) dj(Vi) for all 1 ≤ j ≤ r converges to finite value while (iii) dj(Vi) for all j > r (strictly) diverges to infinity when i tends to infinity. Here, what we meant by the strict divergence in the above (iii) is that all subsequences diverge. We assume the above three conditions throughout the rest of present subsection. Let us first start with the simplest situation that is those satisfying the following conditions. (iv)Xi = 0, Bi = Ig (unit matrix), (v)dj(Vi) = aj for all j ≤ r and (vi) dj(Vi) = i ⋅ aj for all j > r. The real constants a1, ⋯ , ag above satisfy that $$\begin{align*} 1<u_{0}a_{0}, a_{i}<u_{0}a_{i+1}. \end{align*}$$ Intuitively g − r is the corresponding “torus rank” of limit. Then from the above assertions, it is easy to see that Proposition 2.12. The pointed Gromov–Hausdorff limit of the rescaled Kähler–Einstein metrics on |$V_{i}(i\to \infty )$| with fixed injectivity radius 1 in the above notation is isometric to $$\begin{align*} \prod_{r< j\le g} S^{1}\left(\frac{a_{g}}{2\pi a_{j}}\right)\times \mathbb{R}^{g+r},\end{align*}$$ where S1(a) denotes a circle with radius a. Note that “pointed” does not cause ambiguity in this situation, thanks to the homogenuity of abelian varieties. Sketch proof. In the above simple situation, Vi with the Kähler-Einstein metrics on |$V_{i}(i\to \infty )$| is decomposed as $$\begin{align*} \prod_{1\le j\le g}\left(\mathbb{C}/\mathbb{Z}+\mathbb{Z}\sqrt -1 \left(d_{j}(V_{i})\right)^{2}\right)\end{align*}$$ as Kähler manifolds and each |$\mathbb {C}/\mathbb {Z}+\mathbb {Z}\sqrt -1 (d_{j}(V_{i}))^{2}$| is isometric to the product metric space |$S^{1}\left (\frac {1}{2\pi (d_{j}(V_{i}))}\right )\times S^{1}\left (\frac {(d_{j}(V_{i}))^{2}}{2\pi }\right )$|⁠. It concludes that Vi with rescaled Kähler-Einstein metrics on |$V_{i}(i\to \infty )$| with fixed injectivity radius 1 is isometric to $$\begin{align*} \prod_{1\le j\le g}\left(S^{1}\left(\frac{c_{i}}{2\pi d_{j}(V_{i})}\right)\times S^{1} \left(\frac{c_{i} d_{j}(V_{i})}{2\pi}\right)\right),\end{align*}$$ for positive real number ci defined as $$\begin{align*} c_{i}:=\left(\min\left\{\frac{1}{2\pi d_{j}(V_{i})},\frac{d_{j}(V_{i})}{2\pi} \mid 1\le j\le g\right\}\right)^{-1}.\end{align*}$$ Then the assertion of Proposition 2.12 follows. Note that the limit above does not reflect any abelian part data (“a1, ⋯ , ar”) encoded in the boundary of the Satake–Baily–Borel compactification. We prefer the other Gromov–Hausdorff limits, hence we do not pursue the above type rescaled limits further, partially because our main intention is (still) to investigate nice moduli compactifications that occur from other rescalings. 2.5.3 Fixing the volume We remove the assumptions (iv), (v), and (vi) now while keep assuming (i), (ii), and (iii) and analyze the corresponding Gromov–Hausdorff limits while fixing volumes in turn. Note that to fix the volume of {Vi}, say as 1, is simply resulting to the metric matrices $$\begin{align} \left(\begin{array}{cc} Y_{i}^{-1}&Y_{i}^{-1}X_{i}\\ X_{i}Y_{i}^{-1}& X_{i}Y_{i}^{-1}X_{i}+Y_{i}\\ \end{array}\right) \end{align}$$ (5) of |$\mathbb {R}^{2g}/\mathbb {Z}^{2g}$| without any normalization factor. Let B be |$\displaystyle \lim _{i\rightarrow \infty } B_{i}$| and let X be |$\displaystyle \lim _{i\rightarrow \infty } X_{i}$|⁠. We extract the (r × r) upper left part X′ of X and Y ′ of Y as $$\begin{align*} X^{\prime}:= \left(\begin{array}{ccccc} x_{1,1} & x_{1,2}& x_{1,3} & \cdots & x_{1,r} \\ x_{2,1} & x_{2,2} & x_{2,3} & \cdots & x_{2,r} \\ \vdots & \cdots & \cdots & \cdots & \vdots \\ \vdots & \cdots &\cdots & \cdots & \vdots \\ x_{r,1} &x_{r,2} & x_{r,3} & \cdots & x_{r,r} \\ \end{array}\right),\quad Y^{\prime}:=\left( \begin{array}{ccccc} y_{1,1} & y_{1,2} & y_{1,3} & \cdots & y_{1,r} \\ y_{2,1} & y_{2,2} & y_{2,3} & \cdots & y_{2,r} \\ \vdots & \cdots & \cdots & \cdots & \vdots \\ \vdots & \cdots &\cdots & \cdots & \vdots \\ y_{r,1} &y_{r,2} & y_{r,3} & \cdots & y_{r,r} \\ \end{array}\right),\end{align*}$$ and denote the (r × r) upper left part B′ of B as $$\begin{align*} B^{\prime}:=\left(\begin{array}{ccccc} 1 & b_{1,2}& b_{1,3} & \cdots & b_{1,r} \\ & 1 & b_{2,3} & \cdots & b_{2,r} \\ & & 1 & \cdots & \vdots \\ &0 & & \ddots & \vdots \\ & & &&1 \\ \end{array}\right). \end{align*}$$ Then our metric matrices (5) converge to the following except for lower right, that is, (*)-part of size (g − r) × (g − r). Here, the submatrices F, G, and H are those defined by X, X ′, Y, and Y ′ as (Y ′)−1 = F, (Y ′)−1X = G, and X ′(Y ′)−1X ′ + Y ′ = H. The corresponding (*)-part of our metric matrix (5) is exactly the lower right part of Yi, which is diverging due to the divergence of dr+j(Vi) (⁠|$i\rightarrow +\infty $|⁠) for any j > 0. More precisely that (g − r) × (g − r) part is positive definite with all eigenvalues strictly diverge to |$+\infty $|⁠. The diverging part ((g + r + j)-th columns for 1 ≤ j ≤ (g − r)) yields |$\mathbb {R}^{g-r}$| and the rest of part converges to the 2r-dimension real torus with the metric matrix as (6) below. Proposition 2.13. In the above setting, the pointed Gromov–Hausdorff limit of our Vi with fixed volume 1 is isometric to $$\begin{align*} \left(\mathbb{R}^{2r}/\mathbb{Z}^{2r}\right)\times \mathbb{R}^{g-r}\end{align*}$$ where the corresponding metric matrix of the first factor is $$\begin{align} \left(\begin{array}{cc} \textrm{F} &\textrm{G} \\ {}^{t}\textrm{G}& \textrm{H}\\ \end{array}\right). \end{align}$$ (6) The proof follows straightforward from the discussion before the statement. Note that the metric matrix above (6) corresponds exactly to the limit of [Vi]i=1, 2, ⋯ ∈ Ag inside the Satake–Baily–Borel compactification (cf., e.g., [15, 4.4]). In conclusion, we have proved that Corollary 2.14. The Satake–Baily–Borel compactification |$\bar {A_{g}}$|さBB parametrizes the set of pointed Gromov–Hausdorff limits of g-dimensional principally polarized abelian varieties with fixed volumes. This means that the Satake–Baily–Borel compactification [52] can be differential geometrically naturally reconstructed, that is in the spirit of Gromov–Hausdorff. In our sequel with Y. Oshima [51], we further identify |$\bar {A}_{g}^{T}$| with another Satake’s compactification of Ag. 3 Along holomorphic disks In this section, we study the metric behavior of meromorphic polarized family, which means (in this Section 3 of our paper) a flat projective family |$\pi ^{\ast }: (\mathcal {X}^{\ast },\mathcal {L}^{\ast })\to \Delta ^{\ast }$| where |$\Delta ^{\ast }:=\{t\in \mathbb {C}\mid 0<|t|<1\}\subset \Delta :=\{t\in \mathbb {C}\mid |t|<1\},$| which extends to some projective flat polarized family over whole Δ. More precisely, fixing such π*, we take a sequence of points t(i) for i = 1, 2, ⋯ in Δ* converging to the point 0 ∈Δ and consider the Gromov–Hausdorff limit of corresponding metric spaces |$\mathcal {X}_{t(i)}$| for i = 1, 2, ⋯. Of course, it could a priori depends on the sequence t(i) we take, but as a result of the following analysis, it turns out to be not the case! Note that in [50] and our Section 2, we considered all sequential Gromov–Hausdorff limits and hence our task here is to show such independence of the Gromov–Hausdorff limits along a fixed family as π above and specify the subset consists of such “holomorphic limits.” 3.1 Abelian varieties case In this section, we remain on the principally polarized abelian varieties case. Notation 1. This section focuses on the following situation. Take an arbitrary flat projective family of g-dimensional principally polarized abelian varieties over Δ*, which extends to some quasi-projective family over Δ. Replacing by a finite base change if necessary, we can and do assume that it admits (zero-)section, that is, it is a family of algebraic groups and furthermore that we have semi-abelian reduction over 0 ∈Δ by the Grothendieck semi-abelian reduction. We write |$(\mathcal {X}^{\ast },\mathcal {L}^{\ast })\to \Delta ^{\ast }=\Delta \setminus \{0\}$| for such punctured family and the extension as |$(\mathcal {X},\mathcal {L})\to \Delta $|⁠. Set the completion of the local ring of holomorphic functions at 0 as |$R:=\mathbb {C}[[t]]^{conv}$| (the convergent series local ring) and its fraction field |$K:=\mathbb {C}((t))^{mero}$| (the field of meromorphic functions germs at |$t=0\in \mathbb {C}$|⁠). From such germ at 0 of this polarized family, one extracts the following data (“DDample”) as known to [21] (which also at least partially go back to Mumford, Ueno, Nakamura, Namikawa, etc). See [21] for the details. (i) The Raynaud extension |$1\to T\to \tilde {\mathcal {X}}\xrightarrow {\pi } A\to 0$| over R (ii) Ample line bundle |$\tilde {\mathcal {M}}$| on A and |$\tilde {\mathcal {L}}:=\pi ^{\ast }\mathcal {M}$|⁠, (iii) |$X:=\textrm {Hom}(T,\mathbb {G}_{m})$|⁠, |$Y:=\textrm {Hom}(\mathbb {G}_{m},T)$|⁠, the polarization morphism ϕ : Y → X, which is isomorphic in our case. (iv) Y-action on |$\tilde {\mathcal {X}}$| described by |$\iota \colon Y \to \tilde {\mathcal {X}} (K)$|⁠, given by |$\{b(y,\chi )\in \mathcal {O}_{A}\}$| via a (non-unique) isomorphism |$\mathcal {X}\cong \textit {Spec}_{A}(\oplus _{\chi }\mathcal {O}_{A})$|⁠. (v) Set B ′(y, χ) := val(b(y, χ)) and B(y1, y2) := B ′(y1, ϕ(y2)). B is known to be a symmetric positive definite quadric form. Then we analyze the asymptotic behavior of the metrics along this degeneration as follows. Theorem 3.1. For |$(\mathcal {X},\mathcal {L})\to \Delta $| as above, we suppose the extra assumption that the Raynaud extension is the trivial extension (it is satisfied e.g., for the maximally degeneration case). Consider any sequence ti(i = 1, 2, ⋯ ) ∈Δ* converging to 0 then the fiber |$\mathcal {X}_{t(i)}$| with rescaled flat Kähler metric (of diameter 1) |$\frac {d_{\textrm {KE}}\left (\mathcal {X}_{t(i)}\right )}{\textrm {diam}\left (\mathcal {X}_{t(i)}\right )}$| collapses to a r-dimensional real torus where r is the torus rank of |$\mathcal {X}_{0}$| with metric matrix (cB(ei, ej))i, j for a basis {ei}, |$c\in \mathbb {R}_{>0}$| (c is for the rescaling to make the diameter 1). Recall that as we explained at Notation 1, any one parameter family of principally polarized abelian varieties can be reduced to the above form simply by the taking relative Picard space and then some finite base change. Hence the above result in particular confirms a conjecture by Kontsevich–Soibelman [35, Section 5.1, Conjecture 1] for the abelian varieties case and also can be regarded as abelian varieties variant as the conjecture of Gross–Wilson’s [27, Conjecture 6.2] or [25, Conjecture 5.4]. The author heard A. Todorov also had similar conjecture. Proof. By the triviality of the Raynaud extension, |$\mathcal {X}$| is the fiber product over R of a smooth projective family of g-dim principally polarized abelian varieties and another (degenerating) family of principally polarized abelian varieties that has maximal degeneration at 0 ∈ Delta. Then we apply simple [50, Proposition 3.4] and we can easily reduce to the maximally degenerating case, that is, we can assume |$\mathcal {X}_{0}$| is an algebraic torus, without loss of generality. In the maximally degenerating case, we take a uniformizer t of 0 ∈Δ and take an isomorphism |$T\cong \mathbb {G}_{m}^{r}$|⁠, which corresponds to a basis of Y, y1, ⋯ , yr. From the standard way (definition) of the set of data we obtained at Notation 1, the family |$\mathcal {X}_{t}$| with |t|≪ 1 is well known to be written as |$\mathcal {X}_{t}=\mathbb {C}^{r}/M\cdot \mathbb {Z}^{2r}$| where where pi, j(t) is a symmetric matrix with coefficients in the meromorphic functions field |$\mathbb {C}((t))^{ \textit {mero}}\subset \mathbb {C}((t))$|⁠. Indeed, |$p_{i,j}(t)=\frac {1}{b(y_{i},\phi (y_{j}))(t)}$|⁠. Recall B(yi, yj)’s definition from the notation and that it is classically known to be a positive definite matrix. Taking the branch of |$\log (p_{i,j}(t))$| to make its absolute value of the imaginary part at most 2π, the Siegel reduction is automatically done. From our arguments in [50, proofs of 3.1, 3.3], we know that the Gromov–Hausdorff limit of above is determined by the asymptotics of “Y”-(“imaginary”) part of |$\log (p_{i,j}(t))$| for t → 0 that is the orders of pi, j. Hence, |$\mathcal {X}_{t} (t\neq 0)$| converges to |$\mathbb {R}^{r}/\mathbb {Z}^{r}$| with metric matrix B(−, −), appropriately rescaled to make the diameter 1. The author had a discussion on the following phenomenon with A. Macpherson. Corollary 3.2. (“valuative criterion of properness”). Under the assumption of triviality of the Raynaud extension, the Gromov–Hausdorff limit of degenerating abelian varieties |$\mathcal {X}_{t(i)}$| with rescaled flat Kähler metric (of diameter 1) |$\frac {d_{\textrm {KE}}\left (\mathcal {X}_{t(i)}\right )} {\textrm {diam}\left (\mathcal {X}_{t(i)}\right )}$| does not depend on the converging sequences |$t(i)\to 0 (i\to \infty )$|⁠. In other words, the map from Δ* sending t to the underlying metric space of |$\mathcal {X}_{t}$| with the rescaled Kähler–Einstein metric extend to Δ →{compact metric spaces} as a continuous map in the sense of Gromov–Hausdorff. Actually, these 3.1 and 3.2 unconditionally hold for general degeneration of principally degeneration abelian varieties that is without the triviality assumption of the Raynaud extension. This will be proved as a part of joint work with Y. Oshima in a forthcoming paper [51]. Theorem 3.3. (with Y. Oshima [51]). Let |$(\mathcal {X},\mathcal {L})\twoheadrightarrow \Delta $| be as Notation 1. (We do not assume triviality of the Raynaud extension). Consider any sequence {t(i)}i=1, 2, ⋯ ∈Δ* converging to 0 then the fiber |$\mathcal {X}_{t(i)}$| with rescaled flat Kähler metric (of diameter 1) |$\frac {d_{\textrm {KE}}\left (\mathcal {X}_{t(i)}\right )}{\textrm {diam}\left (\mathcal {X}_{t(i)}\right )}$| collapses to a r-dimensional real torus where r is the torus rank of |$\mathcal {X}_{0}$| with metric matrix B appropriately rescaled (to make the diameter 1). In particular, the Gromov–Hausdorff limit of degenerating abelian varieties |$\mathcal {X}_{t(i)}$| with rescaled flat Kähler metric (of diameter 1) |$\frac {d_{\textrm {KE}}\left (\mathcal {X}_{t(i)}\right )}{\textrm {diam}\left (\mathcal {X}_{t(i)}\right )}$| does not depend on the converging sequences |$t(i)\to 0 (i\to \infty )$|⁠. 3.2 Algebraic curves case In this subsection, we analogously study asymptotics of the rescaled Kähler–Einstein metrics of bounded diameters along punctured meromorphic families of compact Riemann surfaces. We do not logically require here the detailed construction of the compactifications of Mg in [50], which is described by the language of the Teichmuller space, its Fenchel–Nielsen coordinates and the pants decompositions. Instead, we give the following brief review of some statements in [50], to provide enough context for our discussion here. The original analog of Theorem 2.3 for compact Riemann surfaces case in [50] was as follows. Theorem 3.4. ([50, Theorem 2.4]). Let {R(i)}i=1, 2, ⋯ be an arbitrary sequence of compact Riemann surfaces of fixed genus g ≥ 2. Suppose |$\left (R(i), \frac {d_{\textrm {KE}}(R(i))} {\textrm {diam(R(i))}}\right )$| (i = 1, 2, ⋯ ) converges in the Gromov–Hausdorff sense. Here dKE denotes the Kähler–Einstein metric on each R(i) and its diameter is diam(R(i)). Then the Gromov–Hausdorff limit is either (i) a metrized (finite) graph of diameter 1 or (ii) a compact Riemann surface of the same genus. Since the Deligne–Mumford compactification |$\bar {M}_{g}^{\textit {DM}}$| with the complex analytic topology is compact, by passing to a subsequence if necessary, we can assume that [R(i)]i=1, 2, ⋯ converges to some |$R(\infty )^{DM} \in \bar {M}_{g}^{ \textit {DM}}$| without loss of generality. Then, the case (i) happens if and only if |$R(\infty )^{DM}$| is non-smooth stable curve and in that case the combinatorial type of the graph is a contraction of the dual graph of the corresponding stable curve |$R(\infty )^{DM}$| that is the limit of [R(i)]i=1, 2, ⋯ in the Deligne–Mumford compactification of the moduli of curves |$\bar {M}_{g}^{ \textit {DM}}$|⁠, with nonnegative metrics (possibly zero) on each edges. Conversely, any metrized dual graph of the stable curve of genus g with diameter 1 can occur as the Gromov–Hausdorff limit in case (i). Corollary 3.5. (cf., [50, Section 2.3, Section 3.2]). $$\begin{align*} \bar{M}_{g}^{T}:=M_{g}\sqcup S_{g}^{wt}\end{align*}$$ with a certain natural topology is a compactification of Mg with complex analytic topology (i.e., a compact Hausdorff topological space, which contains Mg as an open dense subset), where |$S_{g}^{wt}$| denotes the moduli space of metrized finite graph, with nonnegative integer weights w(vi) on each vertex vi, whose underlying topological space Γ satisfies purely combinatorial condition: $$\begin{align} v_{1}(\Gamma)+b_{1}(\Gamma)+\sum_{i}w(v_{i})=g. \end{align}$$ (7) Here, we denote the number of 1-valent vertices as v1(Γ) and denote the first betti number of Γ as b1(Γ). The above condition (7) is nothing but the characterization of finite graphs that can appear as the dual graph of some Deligne–Mumford stable curves of genus g(≥ 2) and the weights encode the genera of the components of the normalization. In the following arguments, we specify which metrized graphs can appear as the Gromov–Hausdorff limits along meromorphic punctured family while also proving that such limits are well defined. We start with setting up the notation of the Kuranishi space of stable curves. 3.2.1 Semi-universal deformations Basic deformation theory of stable curve R tells us that we have a semi-universal (unobstructed) deformation. Its tangent space |$Ext^{1}({\Omega _{R}^{1}},\mathcal {O}_{R})$| maps surjectively to local deformation tangent space |$ \textit {Def}^{\ \textit {loc}}\cong \mathbb {C}^{m}$| whose i-th coordinate corresponds to smoothing one of m nodes xi ∈ R. We first discuss at the semi-universal deformation level in this subsection, and then apply (restrict) that to one parameter deformations later at Section 3.2.2. We anyhow need the Wolpert’s fundamental results in [59] (cf., e.g., also [47]) on asymptotics of the hyperbolic metrics of compact Riemann surfaces along an arbitrary degeneration to a stable curve R. His constructions of smoothing and approximation of the hyperbolic metric are explained as Step 1 and Step 2 below, respectively. We reproduce his results for the convenience of readers and to set up the stage of our later discussions. Step 1. (“plumbing surfaces”). Recall that there is a semi-universal algebraic deformation |$\mathcal {U} \twoheadrightarrow Z$| on an étale cover (variety) Z of |$ \textit {Def}(R)= \textit {Ext}^{1} ({\Omega ^{1}_{R}},\mathcal {O}_{R})$|⁠. We reconstruct its analytic germ in a differential geometric way as follows. We first take an equisingular deformation that is a restriction of |$\mathcal {U}\to Z$| to a closed subset Z ′of Z. This can be also constructed as the product of universal deformation of each components (with nodes marked). We denote this as |$\{R_{s}\}_{s\in Z^{\prime }}$|⁠. We take the normalizations of Rss, which, of course, form a flat family |$R_{s}^{\nu }$| again. The (section formed by) preimages of i-th node(s) (1 ≤ i ≤ m) xi(s) in |$R_{s}^{\nu }$| will be denoted as pi(s) and qi(s). We take a (holomorphic family with respect to s of) local coordinates zi(s), wi(s) around pi(s) and qi(s), respectively, so that $$\begin{align*}z_{i}(s)\left(p_{i}(s)\right)&=0,\\ w_{i}(s)\left(q_{i}(s)\right)&=0.\end{align*}$$ Fix a small enough positive real number c* < 1. Then we construct a small deformation of Rs as $$\begin{align*} R_{s,\vec{t}}:=\left.\left( R_{s}\setminus \left(\bigsqcup_{i}\left\{|z_{i}(s)|<\frac{|t_{i}|}{c_{*}}\right\}\sqcup \left\{|w_{i}(s)|<\frac{|t_{i}|}{c_{*}}\right\}\right)\right)\right/\sim,\end{align*}$$ where for |$\vec {t}=\{t_{i}\}_{i}\in \mathbb {C}^{m}$| with |$|t_{i}|<c_{\ast }^{4}$| and the equivalence relation ∼ is defined on the disjoint union of pairs of sub-annuli $$\begin{align*} \bigsqcup_{i}\left( \left\{\frac{|t_{i}|}{c_{\ast}}\le |z_{i}(s)|\le c_{\ast}\right\}\sqcup \left\{\frac{|t_{i}|}{c_{\ast}}\le |w_{i}(s)|\le c_{\ast} \right\} \right)\end{align*}$$ as $$\begin{align*} z_{i}(s)\sim w_{i}(s) \iff z_{i}(s)w_{i}(s)=t_{i}.\end{align*}$$ Clearly |$R_{s,\vec {t}}$| form a holomorphic flat family of compact Riemann surfaces (equisingular along |$\vec {t}=0$|⁠). We call the image of the annuli in the plumbed Riemann surface |$R_{s,\vec {t}}$| as collars following [59] or sometimes “neck”s in literatures. In this way, we get a flat family |$R_{s,\vec {t}}$| where |$\vec {t}=(t_{1},\cdots ,t_{m})\in \mathbb {C}^{m}$| with |ti|≪ 1, and hence an analytic slice transversal to the equisingular locus Z′ inside the semi-universal deformation space Z. Note that the construction does depend on the local coordinates zi(s), wi(s). Step 2. (“grafting metric”). Next, we set a negative real number a0 < 0 with |a0|≪ 1 (depending on the construction of Step1) fixed and a |$C^{\infty }-$|(“bump”) function |$\eta :\mathbb {R}\to \mathbb {R}$| so that $$\begin{align*} \eta(a)= \begin{cases} 1 & \textrm{if}\ a\le a_{0}\\ \in [0,1] & \textrm{if}\ a_{0}<a<0 \\ =0 & \textrm{if}\ a>0.\end{cases} \end{align*}$$ We call the annuli |$\{\frac {|t_{i}|}{e^{a_{0}}c_{\ast }}<|z_{i}(s)|<e^{a_{0}}c_{\ast }\}$| the collar cores and the complement in the collars that is the set of pairs of annuli |$B_{z_{i}}:=\{e^{a_{0}}c_{\ast }\le |z_{i}(s)|\le c_{\ast }\}$|⁠, |$B_{w_{i}}:=\{e^{a_{0}}c_{\ast }\le |w_{i}(s)|\le c_{\ast }\}$| the collar bands. Then we set a function |$\eta _{z_{i}}$| (resp., |$\eta _{w_{i}}$|⁠) at an open neighborhood of |$B_{z_{i}}$| (resp., |$B_{w_{i}}$|⁠) as $$\begin{align*} \eta_{z_{i}}:=\eta\left(\log \frac{\left|z_{i}(s)\right|}{c^{\ast}}\right)\left(\textrm{resp.,} \eta_{w_{i}}:=\eta\left(\log \frac{\left|w_{i}(s)\right|}{c^{\ast}}\right)\right). \end{align*}$$ Now we “glue” the complete hyperbolic metric on Rs ∖{x1, ⋯ , xm} (i.e., smooth locus), which we denote as |$d{g^{2}_{s}}$| and the local model metric (“with long neck”) |$dg_{loc,t_{i}}^{2}$| around xi, defined as below for |ti|≪ 1. The gluing uses the above bump functions |$\eta _{z_{i}}$| and |$\eta _{w_{i}}$|⁠. Here the local model metric |$dg_{ \textit{loc},t_{i}}^{2}$| is defined as the restriction of $$\begin{align*} \left(\frac{\pi}{\log |t_{i}|}\textrm{csc}\frac{\pi\log |z_{i}(s)|}{\log |t_{i}|}\left|\frac{dz_{i}(s)}{z_{i}}\right|\right)^{2}.\end{align*}$$ In particular, it does not essentially depend on s. Then the actual definition of the smooth (Hermitian) metrics family |$dg^{2}_{s,\vec {t}}$| on |$R_{s,\vec {t}}$| by Wolpert, which he calls grafting, is as follows. $$\begin{align} dg_{s,\vec{t}}^{2}:= \begin{cases} \left(d{g_{s}^{2}}\right)^{1-\eta_{z_{i}}}\cdot \left(dg_{ \textit{loc},t_{i}}^{2}\right)^{\eta_{z_{i}}} &\textrm{around}\ B_{z_{i}}\\ dg_{ \textit{loc},t_{i}}^{2} & \textrm{at the collar core of}\ x_{i}\\ \left(d{g_{s}^{2}}\right)^{1-\eta_{w_{i}}}\cdot \left(dg_{loc,t_{i}}^{2}\right)^{\eta_{w_{i}}} & \textrm{around}\ B_{w_{i}}\\ d{g_{s}^{2}} & \textrm{otherwise}\\ \end{cases} \end{align}$$ (8) The above definition by patching is well defined since at the collar core we have |$\eta _{z_{i}}=\eta _{w_{i}}=1$|⁠. Again, we do the above procedure (Step 2) of grafting the metrics for all nodes xi(i = 1, ⋯ , m) simultaneously. See [59, Section 3], [47, Section 2] for more details if needed. As a result of the above two steps construction, we get a smoothing family of Rs, over a (t1, ⋯ , tm)-polydisc on |$\mathbb {C}^{m}$| which we denote as |$R_{s,\vec {t}}$| and |$C^{\infty }-$|Hermitian metrics family |$dg_{s,\vec {t}}^{2}$| on |$R_{s,\vec {t}}$|⁠. Step 3. Wolpert [59] compares the grafted metrics |$dg_{s,\vec {t}}^{2}$| with the hyperbolic metrics on |$R_{s,\vec {t}}$|⁠. In particular, he proved that they have the “same asymptotic behavior”. From the construction of Step 1 and 2, it is obvious that the grafted metric |$dg_{s,\vec {t}}^{2}$| is the same as local model hyperbolic metric |$ds^{2}_{loc,t_{i}}$| on the collar core with respect to xi that is $$\begin{align*} \left\{\frac{|t_{i}|}{e^{a_{0}}c_{\ast}}<|z_{i}(s)|<e^{a_{0}}c_{\ast}\right\}\end{align*}$$ and coincides with the restriction of the original hyperbolic metric |$d{s_{s}^{2}}$| on R(s). On the collar bands that is $$\begin{align*} \left\{e^{a_{0}}c_{\ast}\le |z_{i}(s)|\le c_{\ast}\right\}\sqcup \left\{e^{a_{0}}c_{\ast}\le |w_{i}(s)|\le c_{\ast}\right\},\end{align*}$$ the grafted metric is a mixture of |$d{g_{s}^{2}}$| and |$dg^{2}_{loc,t_{i}}$|⁠. The crucial result we will use is the following. Fact 3.6. ([59, Lemma 3.5, Section 3.4, 4.2] cf., also [47, p. 690]). The grafted metric |$dg_{s,t}^{2}$| is asymptotically equivalent to hyperbolic metric |$dg_{hyp,s,\vec {t}}^{2}$| in the sense that $$\begin{align*} dg_{s,\vec{t}}^{2}=dg_{hyp,s,\vec{t}}^{2}\cdot \left(1+O\left(\sum_{i}\left(\log|t_{i}|\right)^{-2}\right)\right.,\end{align*}$$ when |$\vec {t}\to 0.$| From the above gluing construction of the metrics by Wolpert, we straightforwardly see that $$\begin{align*} \textrm{diam}\left(i\textrm{-th collar core of}\ \left(R_{s,\vec{t}},\textrm{d}g_{s,\vec{t}}^{2}\right)\right) =\int_{z=\frac{|t_{i}|}{c^{\ast}}}^{c^{\ast}}\frac{\pi}{\log|t_{i}|}\cdot \textrm{csc}\left(\frac{\pi \log|z_{i}|}{\log|t_{i}|}\right)\frac{\textrm{d}z_{i}}{z_{i}}+O(1), \end{align*}$$ for ti → 0, where the last O(1) contribution comes from the |$ \textit {arg}(z(i)) (\in \mathbb {R}/2\pi \mathbb {Z})$|-direction of the collars and the existence of the collarbands. By putting |$|t_{i}|=e^{T_{i}}$|⁠, |$z=e^{T_{i}x_{i}}$| with |$T_{i},x_{i}\in \mathbb {R}$|⁠, the above can be re-expressed as $$\begin{align*} \int_{T_{i}x_{i}=\log(|t_{i}|)}^{T_{i}x_{i}=\log(c^{*})}\frac{\pi}{T_{i}}\textrm{csc}(\pi x_{i})\ \textrm{d}\ (T_{i}x_{i})+O(1). \end{align*}$$ Then a simple calculation shows $$\begin{align*} \int_{T_{i}x_{i}=\log(|t_{i}|)}^{T_{i}x_{i}=\log(c^{\ast})}\frac{\pi}{T_{i}} \textrm{csc}(\pi x_{i})\textrm{d}(T_{i}x_{i})=&\,\int_{T_{i}x_{i}=\log(|t_{i}|)}^{T_{i}x_{i}=\log(c^{\ast})}\textrm{csc}(\pi x_{i})\textrm{d}x_{i} :=\int_{T_{i}x_{i}=\log(|t_{i}|)}^{T_{i}x_{i}=\log(c^{\ast})} \frac{1}{{\sin}(\pi x_{i})}\textrm{d}x_{i}\\ =&\,\int_{T_{i}x_{i}=\log(|t_{i}|)}^{T_{i}x_{i}=\log(c^{\ast})} \frac{\textrm{d}x_{i}}{\pi x_{i}}+O(1)=2\log(-\log(t_{i}))+O(1),\\ \end{align*}$$ for ti → 0. The above calculation is reflecting that, for fixed i, the metrics |$dg^{2}_{loc,t_{i}}$| uniformly converge to |$\left (\dfrac {dz_{i}}{|z_{i}|{\log }|z_{i}|} \right )^{2}$| on any compact subsets of {0 < |z(s)| < c*} when |ti|→ 0. This gives the proof of the Lemma 3.12, when combined with 3.9. If we are allowed to use our slight extension of Morgan–Shalen–Boucksom–Jonsson-type compacfication for more general gluing function (Appendix A.1.3) plus the result of Abramovich et al. [1], we can rephrase and summarize the above outcome into the following statement 3.7. We refer the readers to Appendix A.1.3 and [1] for the details of such preparation and we simply use it without recalling it. However, if one would not have the background and not much interested, then one could possibly skip the following as it essentially just rephrases the above discussions. For those who are interested, please read Appendix (especially Section A.1.3) and [1] as a preparation. Let me only roughly mention that the Morgan–Shalen–Boucksom–Jonsson partial compactification and our extension attach “dual intersection complex” to a given space. Theorem 3.7. There is a natural homeomorphism between our compactification |$\bar {M_{g}}^{T}$| (in [50]) and a variant of Morgan–Shalen–Boucksom–Jonsson compactification of Mg introduced at our Section Appendix A.1.3. That is, $$\begin{align*} \bar{M}_{g}^{T}\cong\bar{M}_{g,(\log(-\log|\cdot|))}^{\textit{hyb}},\end{align*}$$ in the notation of Section A.1.3 in our appendix. More precisely, the above homeomorphism extends the identity of Mg and preserves the corresponding weighted metrized graphs or compact Riemann surfaces’ isomorphism classes when we see the boundary |$\partial \bar {M}_{g,(\log (-\log |\cdot |))}^{ \textit {hyb}}$| as the moduli of such metrized graphs by [1]. Proof of Theorem 3.7. We use the above analysis of (approximation of) hyperbolic metrics based on [59], for the proof. From our construction of Section A.1.3, the boundary of the right-hand side compactification is the dual complex of the boundary of the Deligne–Mumford compactification stack |$\partial \bar {\mathcal {M}_{g}}^{DM}$| that is the quotient of the dual intersection complex of the boundary divisors in charts, divided by the natural equivalence relation induced by the stack structure. We denote such topological space (dual complex) as Δg. On the other hand, Abramovich et al. [1] identified Δg with |$S_{g}^{wt}$| in our notation, preserving the real affine structures on the both sides, thus we have a canonical bijection between the above two spaces extending the identity on Mg. We refer to [1] for more details, if needed. To show that the two topologies, the (generalized) hybrid topology ([10], Appendix A.1.3) and the (weighted) Gromov–Hausdorff topology (cf., [50] and our Subsection 1.1) coincide, it is enough to see that for any given net (in the sense of Bourbaki) {xi∈(Mg⊔Δg)}i∈I converging in both topologies converge to the same point. Since both topologies extend the usual analytic topologies on Mg and also the natural Euclidean topology on the boundary Δg, we can further suppose that all xi are in Mg and it converges to y ∈ Δg for the hybrid topology (resp., z ∈ Δg for the (weighted) Gromov–Hausdorff topology). What we want to show is y = z. Passing to its sub-net if necessary, we can suppose that {xi} converges to a stable curve R in the Deligne–Mumford compactification |$\bar {M_{g}}$| with analytic topology. Then we can lift the net, again by passing to sub-net if necessary, to the level of Defhyb(R) (recall our notation from previous section), which we denote by |$\{\tilde {x_{i}}\}_{i\in I}$|⁠. Then we want to see the equivalence of convergence of |$\{\tilde {x_{i}}\}_{i\in I}$| to some lift |$\tilde {y}$| in Δloc(R) in the hybrid topology and in the (weighted) Gromov–Hausdorff topology. On the other hand, it actually straightforward follows from our proof of Proposition 3.10 since the convergence in the hybrid topology is the convergence of ratio of logarithms of the absolute values of the local coordinates corresponding to the nodes, while it is the same for the weighted Gromov–Hausdorff topology as we analyzed by using [59]. We complete the proof. Remark 3.8. Theorem 3.7 somewhat refines the slogan after Abramovich et al. [1] that “the moduli of skeleton is skeleton of the moduli” (as addressed in A. Macpherson’s talk). For the case of Ag, in our joint work with Y. Oshima [51], we also proved that |$\bar {A_{g}}^{T}$| is identical to Morgan–Shalen–Boucksom–Jonsson compactification of toroidal compactifications of Ag in a slightly extended sense (Appendix A.14). By definition, it in particular proved that the dual complex (in the sense of [1, Section 6], Appendix) of toroidal compactification of Ag does not depend on the choice of cone decompositions and is canonically homeomorphic to our tropical moduli space |$T_{g}=\partial \bar {A_{g}}^{T}$|⁠. This is the abelian variety version of the Abramovich–Caporaso–Payne theorem [1,Theorem 1.2.1]. 3.2.2 One parameter families of curves Now we discuss one parameter family of stable curves. Let us set the notation first. Notation 2. Let |$\mathcal {X}\to \Delta $| be a flat projective family of projective curves over the open unit disk Δ = {|t| < 1}, whose central fiber |$\mathcal {X}_{0}=:R$| is a stable curve. We denote its restriction over the punctured disk Δ*, the smooth projective family as |$\mathcal {X}^{\ast }\to \Delta ^{\ast }$|⁠. Suppose |$\mathcal {X}_{0}$| has nodes x1, ⋯ , xm and around |$x_{i}\in \mathcal {X}$|⁠, we have local equation |$zw=t^{m_{i}}$| with |$A_{m_{i}-1}$|-singularity. The following is also a well known fact and easy to confirm. Fact 3.9. Consider the natural morphism f : Δ →Defloc associated to our family |$\mathcal {X}\to \Delta $|⁠. Then ordt(f*ti) = mi, where ti is a coordinate corresponding to the direction of smoothing xi out. Proposition 3.10. Consider any sequence t(i)(i = 1, 2, ⋯ ) ∈Δ* converging to 0 and suppose that the fiber |$\mathcal {X}_{t(i)}$| with rescaled hyperbolic metric (of diameter 1) |$\frac {d_{\textrm {KE}}\left (\mathcal {X}_{t(i)}\right )}{\textrm {diam}\left (\mathcal {X}_{t(i)}\right )}$| collapses to a metrized finite graph (cf., [50, 2.4]). Then the metrized graph is nothing but the dual graph of |$\mathcal {X}_{0}$| whose edges have all the same lengths. Remark 3.11. We note that the above identification of Gromov–Hausdorff limits for meromorphic family is not sufficient for constructing the compactification itself. Benefiting from the above fact, our proof of 3.10 is reduced to the following. Lemma 3.12. The Gromov–Hausdorff limit of the rescaled grafted metrics on |$\mathcal {X}_{t(i)}$| with the diameter 1 when t(i) → 0 is nothing but the dual graph of |$\mathcal {X}_{0}$| whose edges have all the same lengths. Proof of Proposition 3.10. From our discussion above, it is enough to analyze the diameter contribution of the collars in |$R_{s,\vec {t}}$|⁠. Recall that the grafted metric is nothing but the local model (“long neck”) metric on the collar core while, on the collar bands, the grafted metric lies between the hyperbolic metric on R and the local model, thus its contribution to the diameter is bounded above. Then the assertion follows straightforward from our previous discussions. The above claim especially shows that the Gromov–Hausdorff limit one can get as above in |$\partial \bar {M}_{g}^{T}$| along holomorphic one parameter deformations consist of only finite points! In particular, we also have an analog of Corollary 3.2 for curve case as well. Corollary 3.13. Once we fix the family |$(\mathcal {X}^{\ast },\mathcal {L}^{\ast })\to \Delta ^{\ast }$|⁠, the Gromov–Hausdorff limit of |$\mathcal {X}_{t(i)}$| with rescaled hyperbolic metric (of diameter 1) |$\frac {d_{\textrm {KE}}\left (\mathcal {X}_{t(i)}\right )}{\textrm {diam}\left (\mathcal {X}_{t(i)}\right )}$| for converging sequences ti → 0 |$(i\to \infty )$| do not depend on the choice of the sequence we take and the set of such limits form only a finite subset inside |$\partial \bar {M}_{g}^{T}$|⁠. Remark 3.14. As we briefly introduced at our previous paper [50], L. Lang [37] also introduced the following notion of convergence of compact Riemann surfaces to a metrized finite graph (which he calls tropical curve in the paper), which we analyze and compare with our compactification (cf., also his discussion at [37, v2, Section 1.3]). Definition 3.15. ([37, Definition 1.1]). The expression is somewhat different from the original but the equivalence with it is just a matter of unwinding his definition. A sequence of compact Riemann surfaces of fixed genus g(≥ 2)Ri|i=1, 2, ⋯ converges in the sense of “ tropical convergence” [37] to a metrized finite graph C if the following holds: Ri converges to a stable curve |$R_{\infty }$| in |$\bar {M}_{g}^{DM}$| while shrinking a set of simple geodesics la(Ri) and C underlies a dual graph Γ of |$R_{\infty }$| such that $$\begin{align*} \textrm{length}\left(l_{a}(R_{i})\right)=c_{i}(1+o(1))\frac{1} {\textrm{length}(l_{a}(\Gamma))}\end{align*}$$ for |$i\to \infty $|⁠, for some constants ca, which are independent of i. Here, la(Γ) means the edge of Γ corresponding to the node comes from the shrinking of la(Ri) for |$i\to \infty $|⁠. It directly follows from the known Fact 3.6 (cf., the whole Section 3.2.1 of our review), that his notion of convergence 3.15 is different from our (weighted) Gromov–Hausdorff convergence and actually coincides with the Morgan–Shalen–Boucksom–Jonsson compactification in the slightly extended sense for stacks in the sense of A.1.2. We only sketch the proof as it is quite simple: the approximating grafted metric (8) has the simple closed geodesic of length proportional to |$\frac {1}{|\log (t)|}$| and we can apply such approximation result 3.6 to a finite set of the semi-universal local deformations of stable curves, covering the whole (compact) boundary of the Deligne–Mumford compactification |$\partial \bar {M}_{g}^{DM}$|⁠. Remark 3.16. From the above results, we get a morphism $$\begin{align*} R:M_{g}^{\textit{an}}\to \bar{M}_{g}^{T},\end{align*}$$ where |$M_{g}^{\textit {an}}$| denotes the Berkovich analytification of Mg over the complete discrete valuation field |$\mathbb {C}((t))$|⁠, simply by considering reduction (compare with [1]). The map R is neither continuous nor anti-continuous, in fact it is rather a combination of anti-continuous reduction map over the inner part Mg and continuous tropicalization map over the boundary |$\partial \bar {M}_{g}^{T}$|⁠. Again, from our results in Section 1, we also have a completely analogous map |$A_{g}^{\textit {an}}\to \bar {A}_{g}^{T}$|⁠, which is neither continuous nor anti-continuous. 3.3 Torelli maps It is natural to think how or whether the classical period map $$\begin{align*} t^{\textrm{alg}}: M_{g}\hookrightarrow A_{g}\end{align*}$$ extends between our two compactifications |$\bar {M}_{g}^{T}$| and |$\bar {A}_{g}^{T}$|⁠. Let us briefly recall the recent study of Torelli problem in tropical setting by other mathematicians before. For a unweighted (or weighted with zeroes) metrized graph Γ, the tropical Jacobian [11, 38] is simply |$H_{1}(\Gamma , \mathbb {R}/\mathbb {Z})$| with the following positive definite quadratic form Q. It is defined as $$\begin{align*} Q\left(\sum_{e: \textrm{edge}}\alpha_{e} \cdot e\right):= \sum_{e} {\alpha_{e}^{2}}\cdot l(e)\end{align*}$$ for each 1-cycle |$\sum _{e: \textrm {edge}}\alpha _{e} \cdot e$| where l(−) denotes the length function. Later, this turned out to be equivalent as the skeleton of (generalized) Jacobian in non-Archimedean sense by [7, 58]. By mapping any tropical curve to its tropical Jacobian, [14] and [11] essentially established the existence of a natural map $$\begin{align*} t^{\textrm{Trop}}:\left(S_{g}^{wt}\setminus S_{g}^{wt,tree}\right)\to T_{g}, \end{align*}$$ which is not only continuous but also compatible with their “stacky fan” structure [11] over the cones of these. Here, |$S_{g}^{wt,\textrm {tree}}$| denotes the closed locus of |$S_{g}^{wt}$|⁠, which parametrizes those that underlie trees, that is disjoint from |$S_{g}^{wt,o}$|⁠. (If Γ is a tree, then the tropical Jacobian of [11, 38] is just a point so that we cannot rescale to make the diameter 1.) The Torelli property that is the injectivity of the above does not literally hold even in g = 2 case as pointed out in [38]. Indeed, the closure of only one of the two cells of S2, which parametrizes those without connecting edge, maps onto T2. Nevertheless, they proved that it is “generically one to one” [11, 17]. Now, it is natural to ask the following question of Namikawa–Mumford–Alexeev type (cf., [4, 44]) that is about the extension of talg : Mg↪Ag to compactifications. To be more precise, by combining the above two “period maps” $$\begin{align*} t^{\textrm{alg}}: M_{g}\to A_{g}\end{align*}$$ and $$\begin{align*} t^{\textrm{Trop}}:\left(S_{g}^{wt}\setminus S_{g}^{wt,\textrm{tree}}\right) \rightarrow T_{g}\end{align*}$$ we get a map $$\begin{align*} t_{g}\left(:=t^{\textrm{alg}}\sqcup t^{\textrm{Trop}}\right): {M_{g}^{T}}\setminus S_{g}^{wt,\textrm{tree}}\to \bar{A_{g}}^{T}. \end{align*}$$ Now it is natural to ask the questions of continuity of the map tg. It essentially asks the compactibility of Jacobians and tropical Jacobians. Although we have reviewed this theory of (tropical) Jacobians for the sake of expository completeness, we are afraid that the answer is no! Proposition 3.17. The above map tg is not continuous for any g > 1. Proof. Although we see this failure of continuity in a more systematic way later, we give explicit examples with g = 2 here. We use the fact that usual |$\bar {t}: \bar {M}_{g}^{DM}\to \bar {A}_{g}^{Vor}$| of Mumford–Namikawa [44, Section 18] is isomorphism, which seems to be well known to experts (cf., e.g., [44, Example 18.14]). From more modern perspective, it can be re-explained a little more simpler as follows. First, the pairs of their degenerate abelian varieties and their theta divisors form semi-log-canonical pairs [2, 3.10], [54]. Hence by adjunction, we conclude that such theta divisors are connected nodal curves with ample canonical classes that is stable curves. (Note that, from our modular interpretation, the above isomorphism can be ascended to stacky level |$\bar {\mathcal {M}_{g}}^{DM}\cong \bar {\mathcal {A}_{g}}^{Vor}$|⁠. ) Take a stable curve C0 := C1 ∪ C2 that can be described as follows. Two irreducible components are isomorphic C1≅C2 and are rational curves with one self-intersecting nodal singularity pi(i = 1, 2) each. Furthermore, C1 and C2 intersect transversally at one nodal point q. Take the semi-universal deformation of C0, which is three-dimensional smooth germ with normal crossing discriminant divisors D = D1 ∪ D2 ∪ D ′ components of which are corresponding to local smoothing of pi and q respectively. We take a complex analytic coordinates (z1, z2, z3) corresponding to the divisor D, which satisfy [zi = 0] = Di for i = 1, 2 and [z3 = 0] = D. Consider analytic family of curves {[Ct]=φ(t)∈Mg}t over the unit disk Δ described as |$z_{i}=t^{a_{i}}$| with |$a_{i}\in \mathbb {Z}_{>0}$|⁠. Now, suppose that the map tg is continuous. Then the family φ(t) in M2 converging to a point in |$\partial \bar {M}_{g}^{T}$| corresponding to the graph consists of two circles joined by an edge, which looks like a handcuff. All the three edges have same lengths by Theorem 3.7 for t → 0 (independent of a convergent sequence of t we take). We denote a point corresponding to this metric graph by h. Its tropical Jacobian tg(h) is a 2-dimensional real flat tori S1(c) × S1(c) appropriately rescaled by a positive constant c with the diameter 1. (The exact value of c is |$\sqrt 2$| after all). Here, the value inside the parathesis denote the length of the circumferences of the metrized circle S1. On the other hand, by Theorem 3.1, tg(pi) converges in the Gromov–Hausdorff sense to S1(ca1) × S1(ca3) with appropriate positive constant c to make the diameter 1. Clearly that metric space depends on the parameters ai that contradict to the above. For any g > 2, we can create a counterexample to the continuity of mg as the above counterexample of g = 2 attached with g − 2 elliptic tails. Remark 3.18. If we think of the fact that our compactification |$\bar {M}_{g}^{ \textit {hyb}}$|⁠, to be introduced at our Appendix A.1.2 in much more general context, coincides with Lang's [37]’s compactification, then the existence of continuous map |$\bar {M}_{g}^{ \textit {hyb}}\to \bar {A}_{g}^{T}$| also follows from our later discussion, as a special case of Appendix A.15. For further details, we refer to Appendix A.2 but I hope this to also serve as an introductory motivation for the following appendix. Funding This work was partially supported by the Japan Society for the Promotion of Science [Kakenhi, Grant-in-Aid for Young Scientists (B) 26870316] and [Kakenhi, Grant-in-Aid for Scientific Research (S), No. 16H06335]. Acknowledgments The original version of this preprint was e-print arXiv:1406.7772 appeared in June 2014. Section 2 of this paper is a much revision of the latter half of op.cit. The former half of the original e-print was put as another paper (v2 of arXiv:1406.7772) also with some mathematical and expository improvements. Parts of this work is done when the author visited Chalmers University and Paris. He thanks their warm hospitality and discussions, especially R. Berman, S. Boucksom, C. Favre, M. Jonsson, and A. Macpherson. Since 2014, large part of this paper (and [50], plus some of [51]) has been presented in the talks at various places including Kyoto, Tokyo, Oaxaca, Gothenberg, Oxford, Kanazawa, and Singapore and we appreciate the organizers. We would like to dedicate this set of papers (with [50]) to the heartwarming memory of Kentaro Nagao. We will further continue our series in a forthcoming joint paper with Y. Oshima [51]. I also thank him for his helpful comments to this paper as well. Appendix A Morgan–Shalen type compactification In this appendix, we discuss Morgan–Shalen compactifications [41], in particular, its variants and extensions. We show such compactifying procedure can be naturally extended to fairly general “spaces”, which do not necessarily have good known modular interpretations. The original work [41] was later revisited by DeMarco and McMullen [20], Kiwi [32], and Favre [22] to relate to the Berkovich geometric context and then was partially extended by Boucksom and Jonsson [10] more recently. Our intension of this appendix is to give natural further extensions of [10] (hence [41] partially) to algebraic stacks with some mild singularities allowed and then establish some basic properties. This appendix could be read for independent interest. The main purpose of our extension is that later we use such extensions for our studies of tropical geometric compactifications (cf., e.g., Theorem 3.7 and the remark at the end of Introduction section) by comparison, which will also continue in [51]. In particular, our extensions provide a language to describe our tropical geometric compactifications of Mg, Ag (cf., also [51]). Some part of this appendix requires birational geometric jargon but interested readers who were not accustomed to such language could assume the whole space X to be a smooth orbifold and the boundary divisor X ∖ U = D to be its simple normal crossing divisor. Indeed, it is the most important special case of both our dlt stacky pairs (to be introduced) and toroidal stacks. Indeed, for our later use, such case will be enough. A.1 Slight extensions A.1.1 Brief review of [10, 41] We start with briefly recalling the original constructions of Morgan–Shalen and Boucksom–Jonsson [41], [10, Section 2] in this section. In 1980s, Morgan and Shalen [41] constructed the compactifications of affine complex varieties U = Spec(R) in terms of rings and valuations, depending on the choice of finite generators (as |$\mathbb {C}$|-algebra) of R. We refer to [41, Section 1.3] for the details. They were motivated by studying the character varieties. Recently, Boucksom and Jonsson [10] partially extended the construction to give compactifications of smooth complex varieties U as follows. Starting from algebraic compactification U ⊂ X that is X is a smooth proper variety with D := X ∖ U simple normal crossing divisor, they constructed a “hybrid compactification” of U as follows. We often denote it bravely as |$\bar {U}^{ \textit {hyb}}$| instead of |$\bar {U}^{ \textit {hyb}}(X)$| although it depends on X, in the case if X is obvious from the context. Boucksom and Jonsson [10] wrote this as Xhyb. Set theoretically, the compactification is simply $$\begin{align*} \bar{U}^{ \textit{hyb}}:=U(\mathbb{C}) \sqcup \Delta(D),\end{align*}$$ where Δ(D) denotes the so-called “dual (intersection) complex” (also called the “incidence complex”). See [18, 36] for example. As in [41], they used the logarithmic function to provide compact “hybrid” topology to the above. The topology is characterized by the following: if x ∈ D and local coordinates fi(i = 1, ⋯ , dim(X)) at an open neighborhood satisfying |fi| < 1 for all i, then a sequence xj(j = 1, 2, ⋯ ) of U converging to x, in turn converges in |$\bar {U}^{ \textit {hyb}}$| to a point in Δ(D) with coordinates given by $$\begin{align*} \left(\cdots,\lim_{j\to \infty} \frac{\log\left|f_{i}(x_{j})\right|}{\log\left|\prod_{i} (f_{i}(x_{j})\right|},\cdots\right).\end{align*}$$ We refer to [10, Section 2] for the details. Remark A.1. It is easy to see that the above Boucksom–Jonsson hybrid compactification [10] for a log pair U ⊂ X, where X is smooth and X ∖ U a simple normal crossing divisor, satisfies the same property as our compactifications of Ag and Mg as proved in Corollaries 3.2 and 3.13. That is, if (C ∖{p}) → U is a holomorphic morphism from a punctured disk that extends holomorphically from C to X, the original morphism also extends to a continuous map |$C\to \bar {U}^{ \textit {hyb}}(X)$|⁠. A.1.2 Extending to algebraic stack We work over general algebraically closed field k of characteristic 0 until A.10. We aim at extending the story to the category of algebraic stacks, and for that, we first give a natural set of stacky definitions as a preparation. We start with some obviously natural stacky extension, which has been also naturally discussed in the literatures for other purposes (cf., e.g., [60, Section 4]). In this paper, étale chart of a Deligne–Mumford stack of finite type (over k) means an étale surjective morphism from a locally finite type scheme over k. Definition A.2. A prime divisor of a normal separated Deligne–Mumford stack |$\mathcal {X}$| of finite type over k (DM stack, for short from now on) is a reduced closed substack of |$\mathcal {X}$| of pure codimension 1, which does not decompose as a union of proper closed substacks again of pure codimension 1. Such a divisor |$\mathcal {D}$| is |$\mathbb {Q}$|-Cartier if its pull back to any étale chart is a (algebraically) |$\mathbb {Q}$|-Cartier divisor. It is easy to see that this condition does not depend on the charts. A |$\mathbb {Q}$|-divisor on |$\mathcal {X}$| is a formal |$\mathbb {Q}$|-linear combination of prime divisors |$\mathcal {D}_{1},\cdots ,\mathcal {D}_{s}$| in the form |$\sum _{1\le i\le s}a_{i}\mathcal {D}_{i}$| where all |$a_{i}\in \mathbb {Q}$|⁠. Discussions from the next Definition A.3 until A.7 or A.10 need to assume some acquaintance of the readers with the basic theory of the Minimal Model Program but, for interested readers without it, one might be able to assume that being dlt is only slight extension of simple normal crossing divisors, although very useful, invented by V. Shokurov. Definition A.3. We succeed the above notation. The pair |$\left (\mathcal {X},\sum _{1\le i\le s}a_{i}\mathcal {D}_{i}\right )$| is said to be a stacky log pair if, for any étale chart |$p\colon V\to \mathcal {X}$|⁠, the pair |$\left (V,\sum _{i}a_{i} p^{\ast }\mathcal {D}_{i}\right )$| is a log pair in the sense that |$K_{V}+\sum _{i}a_{i} p^{\ast }\mathcal {D}_{i}$| is |$\mathbb {Q}$|-Cartier. The above pullback |$p^{\ast }\mathcal {D}_{i}$| makes sense since p is étale and it is straightforward to see that this condition does not depend on the presentation p. We remark that by the Keel–Mori theorem [30], we always have a coarse algebraic space X of |$\mathcal {X}$| and its primes divisors Di as coarse subspaces of |$\mathcal {D}_{i}$|s. If we take an étale cover |$V\to \mathcal {X}$| and suppose the natural map V → X branches at prime divisors Bj ⊂ X with order mj, we call the pair |$\left (X,D_{X}:=\sum _{i}a_{i}D_{i}+\sum _{j}\frac {m_{j}-1}{m_{j}}B_{j}\right )$| the “coarse pair” of the stacky log pair |$\left (\mathcal {X},\sum _{i}a_{i}\mathcal {D}_{i}\right )$|⁠. By [33, 5.20] for instance, this |$\left (X,D_{X}\right )$| is also a log pair in the sense the log canonical divisor is |$\mathbb {Q}$|-Cartier. Definition A.4. We succeed the above notation. The stacky log pair |$\left (\mathcal {X},\sum _{1\le i\le s}a_{i}\mathcal {D}_{i}\right )$| is said to be (i) kawamata-log-terminal if (X, DX) is so. (ii) log canonical if (X, DX) is so. Definition A.5. We succeed the above notation. The stacky log pair |$\left (\mathcal {X},\sum _{1\le i\le s}a_{i}\mathcal {D}_{i}\right )$| is said to be locally divisorially-log-terminal or simply dlt stacky pair for bravity, if there is an étale chart |$p\colon V\twoheadrightarrow \mathcal {X}$| with |$(V,\sum _{i}a_{i}p^{*}\mathcal {D}_{i})$| dlt with |$\mathbb {Q}$|-Cartier |$p^{*}\mathcal {D}_{i}$|s. Here the above |$\mathbb {Q}$|-Cartierness again means the algebraic |$\mathbb {Q}$|-Cartierness on given normal variety V. The above notion, extending the (schematic) dlt pair with |$\mathbb {Q}$|-Cartier boundary components, plays a central role in this appendix. Recall that a useful point of the concept of dlt comes from that all the lc centers inside the boundary of dlt pair are generically normal crossings as [23, Section 3.9] shows (cf., also [34, 4.16]). However there is a subtlety that a log pair (in the category of varieties) being dlt stacky pair is not quite the same as dlt pair nor |$\mathbb {Q}$|-factorial dlt pair, first as the condition is only required étale locally and second for the |$\mathbb {Q}$|-Cartierness assumption of the boundaries. The coarse pair of dlt stacky pair around 0-dimensional lc center with |$\mathbb {Q}$|-Cartier boundary components is called “qdlt” (quotient-dlt) in [18]. For our purposes, dual complex of the boundary at the coarse moduli space is not enough and essentially need stack structures as the following simple example shows (cf., also [1, 6.1.7]). Example A.6. Think of the quotient stack |$[(\mathbb {A}_{x,y}^{2},(xy=0))/G]$|⁠, where |$G:=\mathbb {Z}/2\mathbb {Z}$| acts on the affine plane by switching the coordinates that is x↦y, y↦x. Then the dual complex of the quotient is just one point while that of the stack is a segment divided by |$\mathbb {Z}/2\mathbb {Z}$|⁠. (If one would like a compact example, then replace |$\mathbb {A}^{2}$| simply by its projective compactification |$\mathbb {A}^{2}\subset \mathbb {P}^{2}$|⁠.) Definition A.7. We succeed the above notation. A line bundle on a DM stack |$\mathcal {X}$| is said to be nef (resp., ample) if it descends to a nef (resp., ample) |$\mathbb {Q}$|-line bundle on X. Definition A.8. We succeed the above notation. The stacky log pair |$\left (\mathcal {X},\sum _{1\le i\le s}a_{i}\mathcal {D}_{i}\right )$| is said to be (i) stacky klt model if it is stacky klt and |$K_{\mathcal {X}}+\sum _{1\le i\le s}a_{i}\mathcal {D}_{i}$| is nef. (ii) stacky lc model if it is stacky lc and |$K_{\mathcal {X}}+\sum _{1\le i\le s}a_{i}\mathcal {D}_{i}$| is ample. (iii) stacky dlt model if it is dlt and |$K_{\mathcal {X}}+\sum _{1\le i\le s}a_{i}\mathcal {D}_{i}$| is nef. Please do not confuse stacky dlt pair and stacky dlt model (only the latter, which is the special cases of the former, requires the log-minimality condition). We can define the dual complex of any stacky dlt pairs as follows. Definition-Proposition A.9. (Skeleta). For an arbitrary separated stacky dlt pair |$\left (\mathcal {X},\mathcal {D}_{\mathcal {X}}=\sum _{i}\mathcal {D}_{i}\right )$|⁠, we take an étale cover |$p: V\to \mathcal {X}$| and set |$W:=V\times _{\mathcal {X}}V$| with naturally induced morphisms qi(i = 1, 2) : W → V (so that |$\mathcal {X}=[W\rightrightarrows V]$|⁠). Now we consider the colimit of topological spaces |$\Delta (\lfloor (p\circ q_{i})^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor ) \rightrightarrows \Delta (\lfloor p^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor )$|⁠, where i = 1, 2, Δ(−) denotes the dual complex (as in [18]) and the morphisms are affine linear at each simplex which extends the maps of vertices. We denote the colimit as topological space by |$\Delta (\mathcal {D}_{\mathcal {X}})$| and call the dual (intersection) complex or the skeleton of the stacky dlt pair |$(\mathcal {X},\mathcal {D}_{\mathcal {X}})$|⁠. Then, |$\Delta (\mathcal {D}_{\mathcal {X}})$| does not depend on the choice of p and hence well-defined which we call dual complex of the stacky dlt pair |$(\mathcal {X},\mathcal {D}_{\mathcal {X}})$|⁠. Proof. First we untangle the abstract definition of |$\Delta (\mathcal {D}_{\mathcal {X}})$| as a cell complex in more concrete terms. Our topological colimit of |$\Delta (\lfloor (p\circ q_{i})^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor )\rightrightarrows \Delta (\lfloor p^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor )$| has an inductive “skeleton” structure, as being a cell complex, as follows (in the context of cell complexes, rather than that of Berkovich geometry). It is simply because both |$\Delta (\lfloor (p\circ q_{i})^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor )$| and |$\Delta (\lfloor p^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor )$| have stratifications by the (inner parts of) k-skeleta and the two maps between them preserve the stratifications. Now, the 0-skeleton |$\Delta ^{(0)}(\mathcal {D}_{\mathcal {X}})\subset \Delta (\mathcal {D}_{\mathcal {X}})$| is simply the colimit set (with discrete topology) of two maps |$\Delta ^{(0)}(\lfloor (p\circ q_{i})^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor )\rightrightarrows \Delta ^{(0)}(\lfloor p^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor )$|⁠. Above Δ(0)(−) simply denotes the sets of irreducible components (of each divisor −). We then proceed inductively as follows. Suppose we have constructed up to (k − 1)-skeleton part (⁠|$k\in \mathbb {Z}_{>0}$|⁠) of |$\Delta (\mathcal {D}_{\mathcal {X}}),$| which we denote as |$\Delta ^{(k-1)}(\mathcal {D}_{\mathcal {X}})$|⁠. We write the set of codimension k log-canonical centers of |$\lfloor p^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor $| (resp., |$\lfloor (p\circ q_{i})^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor $|⁠) as |$C^{(k)}(\lfloor p^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor )$| (resp., |$C^{(k)}(\lfloor (p\circ q_{i})^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor )$|⁠) and let |$\tilde {\Delta }^{k}(\lfloor p^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor )$| (resp., |$\tilde {\Delta }^{k}(\lfloor (p\circ q_{i})^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor )$|⁠) be defined as $$\begin{align*} \bigsqcup_{S\in C^{(k)}(\lfloor p^{\ast}\mathcal{D}_{\mathcal{X}} \rfloor)} k\textrm{-simplex } \Delta_{S}\left(\textrm{resp., }\bigsqcup_{S\in C^{(k)}\left(\lfloor (p\circ q_{i})^{\ast}\mathcal{D}_{\mathcal{X}}\rfloor\right)} k\textrm{-simplex } \Delta_{S}\right).\end{align*}$$ Then, as the next step, we glue the topological colimit of the induced diagram $$\begin{align*} \tilde{\Delta}^{k}\left(\left\lfloor p^{\ast}\mathcal{D}_{\mathcal{X}} \right\rfloor\right) \rightrightarrows \tilde{\Delta}^{k}\left(\left\lfloor (p\circ q_{i})^{\ast}\mathcal{D}_{\mathcal{X}} \right\rfloor\right)\end{align*}$$ along the natural boundary map |$\partial \tilde {\Delta }^{k} (\lfloor (p\circ q_{i})^{\ast }\mathcal {D}_{\mathcal {X}} \rfloor ) \to \Delta ^{(k-1)}(\mathcal {D}_{\mathcal {X}})$|⁠. Note that the above maps are all cellular. Then we continue up to |$k=\textrm {dim}(\mathcal {X})$| so that the final outcome is nothing but our colimit |$\Delta (\mathcal {D}_{\mathcal {X}})$| using the cover |$V\twoheadrightarrow \mathcal {X}$|⁠. What we want to show is that the above |$\Delta (\mathcal {D}_{\mathcal {X}})$| constructed via the chart V does not depend on the choice of V. Such independence assertion amounts to show the following: if |$[W^{\prime }\rightrightarrows V^{\prime }]$| is another presentation of |$\mathcal {X}$| with an étale morphism f : V ′→ V, then $$\begin{align} \Delta\left(\left\lfloor (p\circ r)^{\ast}\mathcal{D}_{\mathcal{X}} \right\rfloor\right) \twoheadrightarrow\Delta\left(\left\lfloor (p\circ f)^{\ast}\mathcal{D}_{\mathcal{X}} \right\rfloor\right)^{2} \times_{\Delta(\lfloor p^{\ast}\mathcal{D}_{\mathcal{X}} \rfloor) \times \Delta(\lfloor p^{\ast}\mathcal{D}_{\mathcal{X}} \rfloor)} \Delta\left(\left\lfloor (p\circ q_{i})^{\ast}\mathcal{D}_{\mathcal{X}} \right\rfloor\right), \end{align}$$ (A1) that is the above natural morphism is surjective (it is not injective in general, which makes an obstacle to define the dual complex of algebraic stacks at topological stack level for our general setting. See [1, 6.1.9. 6.1.10] for related discussions), where r : (V ′× V ′) ×(V×V)W≅W ′→ V denotes the naturally induced morphism. Also note that since |$q_{1}^{\ast }\mathcal {D}_{\mathcal {X}}=q_{2}^{\ast }\mathcal {D}_{\mathcal {X}}$| as |$\mathcal {D}_{\mathcal {X}}$| is a stacky divisor the right-hand side of (A1) is independent of i. To prove the above required surjectivity (A1) at the level of k-skelta by induction on k is fairly straightforward as follows. First, such assertion for the k = 0 case is surjectivity of the natural map $$\begin{align} C^{(0)}\left(\left\lfloor (p\circ r)^{\ast}\mathcal{D}_{\mathcal{X}} \right\rfloor\right) \twoheadrightarrow C^{(0)}\left(\left\lfloor (p\circ f)^{\ast}\mathcal{D}_{\mathcal{X}} \right\rfloor\right)^{2} \times_{(\Delta^{(0)}(\lfloor p^{\ast}\mathcal{D}_{\mathcal{X}} \rfloor))^{2}} C^{(0)}\left(\left\lfloor (p\circ q_{i})^{\ast}\mathcal{D}_{\mathcal{X}} \right\rfloor\right). \end{align}$$ (A2) This holds immediately as (q1 × q2) : W → V × V is étale to its image and f is also étale. Suppose that we know (A1) up to (k − 1)-skeleta level. Then we want to show that the k-dimensional cells canonically coincides between the both hand sides of (A1). This is nothing but the same claim as (A2) above also holds when we replace 0 by k but, by definition, it is straightforward by the same reason that q1 × q2 and f are étale at open neighborhoods of generic points of codimension k log-canonical centers. The above obviously extends the construction in schematic case (cf., e.g., [18, 46]) and also coincides with [1, Section 6] when overlaps. In particular, note that the above defined dual complex is not the same as the dual complex of the coarse pair. Indeed, the previous Ex. A.6, [1, 6.1.7] provide simple counterexamples. It is natural to expect that roughly speaking the dual complex of “minimal model” does not depend on the choice. More precisely, we conjecture the following after [18], which establish its some versions for schematic case. Conjecture A.10. (Minimal skeleton). Once we fix a (kawamata-)log terminal DM stack |$\mathcal {U}$|⁠, then the homeomorphic type of the dual complex of stacky dlt model |$(\mathcal {X},\sum _{1\le i\le s}\mathcal {D}_{i})$| with |$\mathcal {X}\setminus \sum _{i}\mathcal {D}_{i}=\mathcal {U}$| does not actually depend on the choice of such compactifications. From here, we assume the ground field k is |$\mathbb {C}$|⁠. Definition-Proposition A.11. (Compactifications). We keep the notation of A.9. Then for the open substack |$\mathcal {U}:=\mathcal {X}\setminus \textit {Supp}(\mathcal {D}_{\mathcal {X}})$| and its coarse moduli space U, we can construct a Morgan–Shalen–Boucksom–Jonsson partial compactification |$\bar {U}^{ \textit {hyb}}(\mathcal {X}):=U\sqcup \Delta (\lfloor \mathcal {D}_{\mathcal {X}}\rfloor )$| with a Hausdorff topology extending the complex analytic topology of |$U(\mathbb {C})$|⁠. If |$\mathcal {X}$| is proper and |$\mathcal {D}_{\mathcal {X}}$| is a (effective) |$\mathbb {Z}$|-divisor, then |$\bar {U}^{ \textit {hyb}}(\mathcal {X})$| is also compact. Proof. We simply imitate the construction of Boucksom and Jonsson [10], which we reviewed at A.1.1. We write for the preimage of U to V (resp., W) as UV (resp., UW). Then we first construct |$\bar {U_{V}}^{ \textit {hyb}}(V)$| essentially following the method of [10, Section 2] as follows. For each point x ∈ V ∖ UV, consider the log-canonical center Z, which includes x. By the definition of stacky dlt pair, if we replace V by sufficiently small open subset Vx of V, we can and do assume such log-canonical center is the intersection of m|$\mathbb {Q}$|-Cartier boundary divisor |$D_{V_{x},i} (i=1,\cdots ,m)$| and for sufficiently divisible li ≫ 0, liDV, i is Cartier so that they can be written as fi = 0 by some holomorphic fi on Vx. We shrink Vx small enough to the locus |fi| < 1 if necessary. By running all such x and using these fis on each Vx, we can construct the partial compactification of Vx as |$\overline {\left (V_{x}\setminus (\cup _{x,i}D_{V_{x},i})\right )}^{ \textit {hyb}}(V_{x})$| completely similarly as [10,§2.2]. Then, from our constructions, |$\{\overline {(V_{x}\setminus (\cup _{x,i}D_{V_{x},i}))}^{ \textit {hyb}}(V_{x})\}_{x}$| naturally glue together to form |$\bar {U_{V}}^{ \textit {hyb}}(V)$|⁠. In the same way, we also get |$\bar {U_{W}}^{ \textit {hyb}}(W)$|⁠. Now, we construct |$\bar {U}^{ \textit {hyb}}(\mathcal {X})$| as the (topological) colimit of the natural diagram |$\bar {U_{W}}^{ \textit {hyb}}(W)\rightrightarrows \bar {U_{V}}^{ \textit {hyb}}(V)$|⁠. Independence of such colimit from the cover V is proved completely similarly as the above proof of A.9, thus we omit the details. Example A.12. (Torus embedding case description). We give a description for toric case. What we mean is the following. Starting from arbitrarily proper torus embedding T ⊂ X, we can do toric log resolution of (X, X ∖ T); |$f\colon \tilde {X}\to X$|⁠. Then with trivial stack structure, we can talk about the dual complex of |$\tilde {X}\setminus T$| and corresponding Morgan–Shalen–Boucksom–Jonsson compactification of |$T(\mathbb {C})$|⁠. Then we can concretely see the resulting compactifications indeed do not depend on the (complete) fan structure. See [31] for the basics of the toric (or toroidal) geometry. Let |$N\cong \mathbb {Z}^{n}$| be a lattice and |$T:=T_{N}:=N\otimes _{\mathbb {Z}}\mathbb {G}_{m}$| be the associated algebraic torus. We take a basis of N so that we sometimes identify N as |$\mathbb {Z}^{n}$| and T as |$(\mathbb {G}_{m})^{n}$|⁠. We consider the tropicalization map (it can be also seen as the moment map with respect to the (S1)n(⊂ T)-action and the Kähler form |$\prod _{i} \frac {dz_{i}\wedge {\bar dz_{i}}}{|z_{i}|^{2}}.$|⁠) $$\begin{align*} m: T(\mathbb{C}) \twoheadrightarrow N_{\mathbb{R}}:=N\otimes_{\mathbb{Z}}\mathbb{R},\end{align*}$$ which is, via the basis of N, written as $$\begin{align*} (z_{1},\cdots,z_{n})\mapsto (-\log |z_{1}|,-\log |z_{2}|,\cdots,-\log |z_{n}|).\end{align*}$$ It is easy to verify that this definition does not depend on the choice of basis of N. This logarithmic mapping is a key in the construction of [10]. It is natural to attach the infinite hyperplane to form the natural projective compactification |$N_{\mathbb {R}}\subset \mathbb {P}_{N_{\mathbb {R}}}=:\mathbb {P}=N_{\mathbb {R}}\sqcup ((N_{\mathbb {R}}\setminus \{0\})/\mathbb {R}^{\ast })$|⁠. We also denote the boundary |$((N_{\mathbb {R}}\setminus \{0\})/\mathbb {R}^{\ast })$| as |$\partial \mathbb {P}$|⁠. Note this compactification is different from |$N_{\mathbb {R}}\cong \mathbb {R}^{n}\subset (\mathbb {R}\sqcup \{+\infty \})^{n}$| relative to the interior. From the compactness of the real projective space |$\mathbb {P}$|⁠, for any sequence xi(i = 1, 2, ⋯ ) ∈ T, after passing to subsequence if necessary, m(xi) converge to some point in |$\mathbb {P}$|⁠. Then, the natural hybrid compactification for a torus embedding T ⊂ X with simple normal crossing toric boundary X ∖ T by [10] can be reconstructed as $$\begin{align*} T\sqcup \left((N_{\mathbb{R}}\setminus \{0\})/\mathbb{R}^{\ast}\right).\end{align*}$$ The natural topology we put on the above space as an extension of the complex analytic topology on T is defined by the convergence of sequences (or nets) of points xi of T, which does not converge inside T, to a point of boundary |$\partial \mathbb {P}$| as the convergence of the sequence m(xi). It is easy to see that this is equivalent to the definition of [10] (cf., also A.1.1), thus independent of the choice of above toric compactification. Here is the local description. For an affine toric variety |$(T\subset ) X=U_{\sigma }= \textit {Spec}(\mathcal {S}_{\sigma })$|⁠, we simultaneously imitate and extend the above. Each |$m\in \mathcal {S}_{\sigma }$| corresponds to a regular function e(m) on Uσ of monomial type and we take a finite subset |$S\subset \mathcal {S}_{\sigma }$|⁠, which generates the function ring |$\mathcal {S}_{\sigma }$|⁠. Then we define the moment map $$\begin{align*} m_{S}: X\to \mathbb{R}^{S}\end{align*}$$ as |$x\mapsto (-\log |e(m)(x)|)_{m\in S}$|⁠. It is standard to see that this is nothing but the combination of surjection |$X\twoheadrightarrow X/CT$|⁠, where CT := N ⊗ U(1) is the natural compact form of T, followed by the well-known topological embedding of X/CT into a manifold with corners (cf., [48]). Then finally we consider $$\begin{align*} \partial^{\sigma} X:=\left\{\lim_{i\to +\infty}m_{S}(x_{i})\in ((N_{\mathbb{R}}\setminus \{0\})/\mathbb{R}^{\ast}) \mid x_{i}\ \textrm{converges in}\ U_{\sigma}\right\},\end{align*}$$ and set |$\bar {X}^{\sigma }:=X\sqcup \partial ^{\sigma } X$| with natural topology defined by the above convergence. It follows straightforward from the construction that X is open dense inside |$\bar {X}^{\sigma }$|⁠. It is also easy to see that the partial compactification |$X\subset \bar {X}^{\sigma }$| does not depend on the choice of S and furthermore that as far as S with the origin spans the maximal (n-) dimensional space, the above construction works and gives the same outcome |$\bar {X}^{\sigma }$|⁠. Our construction of this |$\bar {X}^{\sigma }$| is a special case of [41, Section I.3]. Example A.13. (For toroidal stacks [1]). Here we follow [1] and [55]. Although not all toroidal DM stack (cf., [1, 6.1.1]) form dlt stacky pairs as not all toric singularities are dlt, [1, 55] nevertheless give a natural partial generalization of the dual complex construction to toroidal embedding stack|$\mathcal {U}\subset \mathcal {X}$| to form |$\Delta (\mathcal {X}\setminus \mathcal {U})$| (especially [1, Section 6]) extending the skeleton of Thuillier [55]. Note that this construction [1, 55] essentially uses the natural log structure and indeed it is further extended to general fine saturated log schemes by [57]. Let us briefly review the construction and give a corresponding Morgan–Shalen-type compactification from a perspective of the previous discussion A.12. The resulting compactification will be denoted as |$\bar {U}^{ \textit {hyb}}:=U\sqcup (\Delta (\mathcal {X}\setminus \mathcal {U}))$|⁠, where U is the coarse moduli space of |$\mathcal {U}$|⁠. Let us take an étale cover |$p\colon V\to \mathcal {X}$|⁠, denote the preimage of |$\mathcal {U}$| as UV, and set |$W:=V\times _{\mathcal {X}}V$| as before. Then we put DV := V ∖ UV. For each point x ∈ DV, we can take a Euclidean open neighborhood of x ∈ UV, x ⊂ V, which has analytic isomorphism to a Euclidean open neighborhood of a point in the boundary of some affine toric variety Vx. We suppose that all the boundary components intersect the open subset UV, x and denote the isomorphism as ix : UV, x↪Vx. Denote the corresponding cone of Vx by σx and set the dual |$\mathcal {S}_{\sigma _{x}}:=\{m\in M:= \textit {Hom}_{\mathbb {Z}}(N,\mathbb {Z})\mid (m,n)\ge 0 \textrm { for all }n\in \sigma _{x} \}$| so that |$V_{x}= \textit {Spec}(\mathbb {C}[\mathcal {S}_{\sigma _{p}}])$|⁠. Each m ∈ M corresponds to a function e(m) on Xx so |$i_{x}^{\ast }(e(m))$| on UV, x. Take a finite generating system {mi}i∈S of the semigroup |$\mathcal {S}_{\sigma _{x}}$| and exploits the partial compactification in the previous section, that is, we consider |$\partial ^{\sigma _{x}} X_{x}$| and correspondingly we take partial compactification of Vx, which we denote by |$\bar {V}^{\{x\}}$|⁠. It is straightforward to see that |$\bar {V}^{\{x\}}$| glues together to form a partial compactification |$\bar {V}$|⁠. Indeed, if we take another isomorphism |$i_{p}^{\prime }: U_{x}\cong V_{x}\subset X_{x}$|⁠, by shrinking Ux if necessary, |$\frac {(i_{x}^{\prime })^{\ast }(e(m))}{i_{x}^{\ast }( e(m))}$| are non-vanishing well-defined function on Ux for any m ∈ M (we can check this by restricting to finite generators). Similarly we can do the same construction to form a partial compactification |$\bar {W}$| of W. As in the previous A.11 we define the desired generalized hybrid compactification of U as the colimit of |$\bar {W}\rightrightarrows \bar {U}$|⁠. Independence of the construction from V is proved completely similarly as A.11 so we avoid to repeat the details of its proof. Remark A.14. ([51]). For toroidal compactifications of locally Hermitian symmetric space [6], we can also naturally assign hybrid compactification as either special case of Definition-Proposition A.11 (when it is smooth stack with normal crossing boundary) or Example A.13 in general. It is straightforward from the constructions that it does not depend on the admissible cone decompositions. Then, in a forthcoming paper [51], we showed that first it does not depend on the admissible cone decompositions and such compactification for Ag case coincides with our |$\bar {A_{g}}^{T}$|⁠. A.1.3 About “gluing function” This subsection means to be a simple remark that the logarithmic function used in [10] for the hybrid compactification can be replaced by more general diverging function f, which we would call gluing function. Here is the condition for such functions to be used: |$f: D^{\ast }(\epsilon )\to \mathbb {R}_{>0}$| is a continuous function from |$D^{\ast }(\epsilon ):=\{z\in \mathbb {C} \mid 0<|z|<\epsilon \}$| for 0 < ϵ ≪ 1 such that (i) |$f(z)\to +\infty $| when z → 0, (ii) for any |$c\in \mathbb {C}^{\ast }$|f(cz) − f(z) = O(1) when z → 0. The above condition morally tells that the function grows not (asymptotically) faster than the logarithmic function. Indeed, it is straightforward to see that all our constructions of Morgan–Shalen–Boucksom–Jonsson partial compactifications only use the above properties. Thus, our generalized hybrid compactification is similarly defined as $$\begin{align*} \bar{U}^{ \textit{hyb}}_{(f)}=\bar{U}^{ \textit{hyb}}_{(f(z))}:= U(\mathbb{C})\sqcup \Delta(D),\end{align*}$$ that is set-theoricially same as Boucksom-Jonsson hybrid space, with modified hybrid topology depending on f, but defined just by imitating [10], which was the case |$f(z)=-\log |z|$|⁠. Hence, |$\bar {U}^{ \textit {hyb}}_{(\log |z|)}=\bar {U}^{ \textit {hyb}}$| and so we tend to omit the subscript (f) when f is the usual logarithmic function as above. Note that in our proof of Proposition AppendixA.11, the logarithmic function can be replaced by any function satisfying above so that we can define |$(U\subset )\bar {U}^{ \textit {hyb}}_{(f)}$| for stacky dlt pair |$(\mathcal {X},\sum _{i}\mathcal {D}_{i})$|⁠. A.2 Functoriality of MSBJ construction Theorem A.15. (Functoriality). The skeleta and the Morgan–Shalen–Boucksom–Jonsson partial compactifications ([41],[10], A.9, A.11, A.13, A.1.3) are both functorial in the following sense. Suppose |$(\mathcal {X},\mathcal {D}_{\mathcal {X}})$| and |$(\mathcal {Y},\mathcal {D}_{\mathcal {Y}})$| are dlt stacky pairs (resp., |$(\mathcal {X}\setminus \mathcal {D}_{\mathcal {X}})\subset \mathcal {X}, (\mathcal {Y}\setminus \mathcal {D}_{\mathcal {Y}})\subset \mathcal {Y}$| are toroidal DM stacks) as A.9 (i.e., components of both boundaries have all coefficients 1). If |$f:\mathcal {X}\to \mathcal {Y}$| is a representable morphism (resp., toroidal morphism) such that |$f^{\ast }\mathcal {D}_{\mathcal {Y}}= \mathcal {D}_{\mathcal {X}}$|⁠. Denote the coarse moduli of |$\mathcal {X}\setminus \mathcal {D}_{\mathcal {X}}$| (resp., |$\mathcal {Y}\setminus \mathcal {D}_{\mathcal {Y}}$|⁠) by |$U_{\mathcal {X}}$| (resp., |$U_{\mathcal {Y}}$|⁠). Then the induced map |$U_{\mathcal {X}}\to U_{\mathcal {Y}}$| with complex analytic topologies continuously extends in a unique way to |$\bar {U}_{\mathcal {X}}^{ \textit {hyb}}\to \bar {U_{\mathcal {Y}}}^{ \textit {hyb}}$|⁠. Moreover, if |$(\mathcal {X},\mathcal {D}_{\mathcal {X}})$| and |$(\mathcal {Y},\mathcal {D}_{\mathcal {Y}})$| are both dlt and toroidal, with f toroidal morphism, then the two constructions coincide. For this Theorem A.15, the gluing function needs to be the usual logarithmic function. Proof. First we see that dual complexes are functorial. Passing to an étale cover, we can and do assume |$\mathcal {X}$| and |$\mathcal {Y}$| are varieties. (The necessary arguments for such reduction is again in the same way as our Section A.1.2 so we omit the details.) We create a natural map at the k-skeleta level of |$\Delta (\mathcal {D}_{\mathcal {X}})$| and |$\Delta (\mathcal {D}_{\mathcal {Y}})$| on induction on k. The k-simplices forming |$\Delta (\mathcal {D}_{\mathcal {X}})$| corresponds to codimension k lc centers of |$(\mathcal {X},\mathcal {D}_{\mathcal {X}})$|⁠. If we take a general point x of an arbitrary lc center Z and take a sufficiently small Euclidean open neighborhood Ox of x, Z ⊂ Ox can be written as (z1 = ⋯ = zk = 0) where zi(i = 1, ⋯ , dim(X)) are local holomorphic functions of Ox. If we suppose that the lc center of |$(\mathcal {Y},\mathcal {D}_{\mathcal {Y}})$| is l-dimensional Z ′, we can take local holomorphic functions around f(x) as w1, ⋯ , wdim(Y) with |$(\prod _{1\le i\le l} w_{i}=0)\cap U_{\mathcal {Y}}=Z^{\prime}$|⁠. From the assumptions, we can write f as |$w_{i}=\prod _{1\le j\le k}z_{j}^{m_{i,j}}\cdot g_{i,j}(\vec {z})$| for all i = 1, ⋯ , l with some invertible functions gi, j. (It morally says that f is not so far from monomial maps.) Furthermore, as |$f^{\ast }\mathcal {D}_{\mathcal {Y}}=\mathcal {D}_{\mathcal {X}}$|⁠, |$\sum _{i}m_{i,j}>0$| for any j and |$\sum _{j}m_{i,j}>0$| for any i, hence k ≥ l in particular. The matrix (mi, j)i, j induces a morphism from the k-simplex corresponding to Z to a l-simplex corresponding to Z ′. It naturally glues to form a continuous map |$\Delta (\mathcal {D}_{\mathcal {X}})\to \Delta (\mathcal {D}_{\mathcal {Y}})$|⁠, which does not depend on the choice of |$(U_{\mathcal {X}},z_{j})$| and wis. From the above construction, it is also obvious that the map |$U_{\mathcal {X}}\sqcup \Delta (\mathcal {D}_{\mathcal {X}})\to U_{\mathcal {Y}}\sqcup \Delta (\mathcal {D}_{\mathcal {Y}})$|⁠, which is simply obtained as a disjoint union of the two maps, is continuous. □ By applying the above Theorem A.15 to the extended Torelli maps |$\bar {\mathcal {M}_{g}}\to \bar {\mathcal {A}_{g}}$| ([4, 45], etc.) we will get more counterexamples systematically to the continuity of the “glued” Torelli map tg we discussed around Theorem 3.17. References [1] Abramovich , D. , L. Caporaso , and S. Payne . “ The tropicalization of the moduli space of curves .” Ann. Sci. Éc. Norm. Supér. , no. 4 ( 2015 ). WorldCat [2] Alexeev , V. “Log canonical singularities and complete moduli of stable pairs.” arXiv:9608013. [3] Alexeev , V. “ Complete moduli in the presence of semiabelian group action .” Ann. of Math. 2 , ( 2002 ): 611 -- 708 . Google Scholar Crossref Search ADS WorldCat [4] Alexeev , V. “ Compactified Jacobians and Torelli map .” Publ. Res. Inst. Math. Sci. 40 , ( 2004 ): 1241 -- 65 . Google Scholar Crossref Search ADS WorldCat [5] Alexeev , V. , and I. Nakamura . “ On Mumford’s construction of degenerating abelian varieties, Tohoku .” Math. J. (2) 51 , ( 1999 ): 399 -- 420 . Google Scholar Crossref Search ADS WorldCat [6] Ash , A. , D. Mumford , M. Rapoport , and T.-S. Tai . Smooth Compactifications of Locally Symmetric Varieties . 2nd ed. Cambridge Mathematical Library, 2010 . Google Preview WorldCat COPAC [7] Baker , M. , and J. Rabinoff . “ The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves ”. Int. Math. Res. Not . 16 ( 2015 ): 7436 -- 72 . Google Scholar Crossref Search ADS WorldCat [8] Borel , A. “ Stable real cohomology of arithmetic groups .” Ann. Sci. Éc. Norm. Supér. 4 , ( 1974 ): 235 -- 72 . Google Scholar Crossref Search ADS WorldCat [9] Borisov , L. , L. Chen , and G. Smith . “ The orbifold Chow ring of toric Deligne-Mumford stacks .” J. Amer. Math. Soc. 18 ( 2005 ), no. 1 , 193 -- 215 . Google Scholar Crossref Search ADS WorldCat [10] Boucksom , S. , and M. Jonsson . “ Tropical and non-Archimedean limits of degenerating families of volume forms .” to appear in JÉc. polytech. Math. WorldCat [11] Brannetti , S. , M. Melo , and F. Viviani . “ On the tropical Torelli map .” Adv. Math. 226 ( 2011 ), no. 3 , 2546 -- 2586 . Google Scholar Crossref Search ADS WorldCat [12] Burago , D. , Y. Burago , and S. Ivanov . A course in Metric Geometry . Graduate Studies in Mathematics, AMS, vol. 33 , 2001 . Google Preview WorldCat COPAC [13] Caporaso , L. “Algebraic and tropical curves: comparing their moduli spaces”. arXiv: 1101 . 4821 ( 2011 ). [14] Caporaso , L. , and F. Viviani . “ Torelli theorem for graphs and tropical curves .” Duke Math. J. 153 , no. 1 ( 2010 ): 129 -- 71 . Google Scholar Crossref Search ADS WorldCat [15] Chai , C.-L. “ Siegel Moduli schemes and their compactification over |$\mathbb {C.}$|” . Chap. IX in Arithmetic Geometry edited by G. Cornell , J H. Silverman, Springer ( 1986 ). [16] Charney , R. “ A generalization of a theorem of Vogtmann ”. J. Pure Appl. Alg . 44 ( 1987 ): 107 -- 25 . Google Scholar Crossref Search ADS WorldCat [17] Culler , M. , and K. Vogtmann . “ Moduli of graphs and automorphisms of free groups ”. Invent. Math . 84 , no. 1 ( 1986 ): 91 -- 119 . Google Scholar Crossref Search ADS WorldCat [18] de Fernex , T. , J. Kollár, and C. Xu . “ The dual complex of singularities ”. Adv. Stud. Pure Math. , Professor Kawamata’s 60th birthday volume. [19] Deligne , P. , and D. Mumford , “The irreducibility of moduli of curves”, Inst. Hautes Études Sci. Publ. Math. no. 36 ( 1969 ): 75 -- 109 . Google Preview WorldCat COPAC [20] DeMarco , L.G. , and C.T. McMullen . “ Trees and the dynamics of polynomials ”. Ann. Sci. Éc. Norm. Supér. ( 2008 ). WorldCat [21] Faltings , G. , and C-L. Chai . Degeneration of Abelian Varieties . Springer. 1990 . Google Preview WorldCat COPAC [22] Favre, C. unpublished notes. dated December, 2012. Cf., also his recent arXiv:1611.08490. [23] Fujino , O. “What is log terminal?” a chapter in Flips for 3-Folds and 4-Folds, Oxford University Press , 2007 . [24] Gromov , M. “ Structures métriques pour les variétés riemanniennes ”. Textes Mathématiques, Paris , no. 1 ( 1981 ): 1 -- 120 . WorldCat [25] Gross , M. “ Mirror Symmetry and the Strominger-Yau-Zaslow conjecture ”. arXiv : 1212 . 4220 ( 2012 ). WorldCat [26] Gross , M. , and B. Siebert . “ Theta functions and Mirror symmetry ”. JDG conference proceeding, arXiv : 1204 . 1991 . WorldCat [27] Gross , M. , and P. M. H. Wilson . “ Large complex structure limits of K3 surfaces ”. J. Differential Geom. 55 , no. 3 ( 2000 ) : 475 -- 546 . Google Scholar Crossref Search ADS WorldCat [28] Iwanari , I. “ The category of toric stacks ”. Compos. Math. 145 ( 2009 ), no. 3 , 718 -- 746 . Google Scholar Crossref Search ADS WorldCat [29] Ji , L. , and J. Jost . “Universal moduli spaces of Riemann surfaces” . arXiv: 1611 . 08732 . [30] Keel , S. , and S. Mori . “ Quotiens by groupoids ”. Ann. of Math. 145 ( 1997 ), no. 1, 193 -- 213 . Google Scholar Crossref Search ADS WorldCat [31] Kempf , G. , F. Knudsen , D. Mumford , and B. Saint-Donat . “ Toroidal embeddings. I ”. Lecture Notes in Mathematics, vol. 339. Springer , 1973 . Google Preview WorldCat COPAC [32] Kiwi , J. “ Puiseux series polynomial dynamics and iteration of complex cubic polynomials ”. Ann. Inst. Fourier ( 2006 ). WorldCat [33] Kollár , J. , and S. Mori . Birational Geometry of Algebraic Varieties , Cambridge Tracts of Mathematics , Cambridge University Press , 1998 . Google Preview WorldCat COPAC [34] Kollár , J. Singularities of the Minimal Model Program, Cambridge Tracts in Mathematics ( 2013 ). [35] Kontsevich , M. , and Y. Soibelman . “ Affine structures and non-archimedian geometry ”. The Unity of Mathematics Progress in Mathematics 244 ( 2006 ) : 321 -- 385 . Google Scholar Crossref Search ADS WorldCat [36] Kulikov , V. “ Degenerations of K3 surfaces and Enriques surfaces ”. Izv. Akad. Nauk S.S.S.R Ser. Mat. 11 , no. 5 ( 1977 ). WorldCat [37] Lang , L. “ Harmonic tropical curves ”. arXiv:1501.07121v2. [38] Mikhalkin , G. , and I. Zharkov . “ Tropical curves, their Jacobians and Theta functions ”. Contemporary Mathematics vol. 465 , Proceedings of the International Conference on Curves and Abelian Varieties in honor of Roy Smith’s 65th birthday , pp. 203 -- 231 ( 2007 ). WorldCat [39] Mirzaii , B. , and W. Van der Kallen . “Homology stability for symplectic groups”. arXiv:0110163 ( 2001 ). COPAC [40] Morgan , J. W. , and Shalen , P. B. . Valuations, trees, and degenerations of hyperbolic structures. I. Ann. of Math. (2) 120 ( 1984 ), no. 3 , 401 – 476 . Google Scholar Crossref Search ADS WorldCat [41] Morgan , J. , P.B. Shalen , “ Valuations, trees and degeneration of hyperbolic structures ”, Ann. of Math. , 122 ( 1985 ). WorldCat [42] Nakamura , I. “ Stability of degenerate abelian varieties ”. Invent. Math . 136 ( 1999 ): 659 -- 715 . Google Scholar Crossref Search ADS WorldCat [43] Nakamura , I. “ Another canonical compactification of the moduli space of abelian varieties, Algebraic and arithmetic structures of moduli spaces (Sapporo, 2007) .” Adv. Studies Pure Math. 58 ( 2010 ): 69 - 135 . WorldCat [44] Namikawa , Y. “ A new compactification of the Siegel space and degenerations of abelian varieties, I, II .” Math. Ann. 221 ( 1976 ): 97–141 , 201 -- 41 . Google Scholar Crossref Search ADS WorldCat [45] Namikawa , Y. “ Toroidal Compactification of Siegel Spaces .” Lecture Notes in Mathematics 812 ( 1980 ). WorldCat [46] Nicaise , J. and C. Xu , “ The essential skeleton of a degeneration of algebraic varieties .” Amer. J. Math. 2016 . WorldCat [47] Obitsu , K. and S. Wolpert . “ Grafting hyperbolic metrics and Eisenstein series .” Math. Annalen , 341 ( 2008 ): 685 -- 706 . Google Scholar Crossref Search ADS WorldCat [48] Oda , T. “ Convex bodies and algebraic geometry .” Ergeb. der Math. und ihrer Grenz . Springer ( 1988 ). Google Preview WorldCat COPAC [49] Odaka , Y. A generalization of the Ross-Thomas slope theory . Osaka J. Math. 50 ( 2013 ), no. 1 , 171 -- 185 . WorldCat [50] Odaka , Y. “ Tropical Geometric Compactification of Moduli, I - Mg case - .” arXiv:1406.7772v2 . WorldCat [51] Odaka , Y. and Y. Oshima , in preparation . [52] Satake , I. “ On the compactification of the Siegel space .” J. Indian Math. Soc. ( 1956 ): 259 -- 81 . WorldCat [53] Strominger , A. , S. T. Yau , and E. Zaslow . “ Mirror symmetry is T-duality .” Nuclear Physics B 479 ( 1996 ): 243 -- 59 . Google Scholar Crossref Search ADS WorldCat [54] Tenini , J. “ On the singularities of degenerate abelian varieties .” ( 2014 ). arXiv: 1401. 0516 . [55] Thuillier , A. “ Géométrie toroïdale et géométrie analytique non archimédienne .” Application au type d’homotopie de certains schémas formels Manuscripta Math ( 2007 ). WorldCat [56] Tyomkin , I. “ Tropical geometry and correspondence theorems via toric stacks .” Math. Ann. 353 , no. 3 ( 2012 ): 945 -- 95 . Google Scholar Crossref Search ADS WorldCat [57] Ulirsch , M. “Functorial tropicalization of logarithmic schemes: the case of constant coefficients.” ( 2013 ). arXiv: 1310. 6269 . [58] Viviani , F. , “ Tropicalizing vs vompactifying the Torelli morphism ,” Contemp. Math. 605 ( 2013 ): 181 -- 210 . arXiv: 1204. 3875 . Google Scholar Crossref Search ADS WorldCat [59] Wolpert , S. , “ The hyperbolic metric and the geometry of the universal curve .” J. Diff. Geom. vol. 31 , no. 2 ( 1990 ): 417 -- 72 . Google Scholar Crossref Search ADS WorldCat [60] Yasuda , T. , “ Motivic integration over Deligne-Mumford stacks .” Adv. in Math. ( 2006 ) arXiv: 0312115 . WorldCat [61] Zhang , Y. “ Collapsing of negative Kähler-Einstein metrics .” arXiv: 1505. 04728 . WorldCat © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com. 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TI - Tropical Geometric Compactification of Moduli, II: Ag Case and Holomorphic Limits
JF - International Mathematics Research Notices
DO - 10.1093/imrn/rnx293
DA - 2019-11-05
UR - https://www.deepdyve.com/lp/oxford-university-press/tropical-geometric-compactification-of-moduli-ii-ag-case-and-2J0ctB7DQc
SP - 6614
VL - 2019
IS - 21
DP - DeepDyve
ER -