In an attempt to extract limiting information from the SYZ picture, Kontsevich and Soibelman [48][49] proposed the following picture. Let $X_{t}$ be a polarized degeneration of Calabi-Yau manifolds near the large complex structure limit, then

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  (Kähler class normalization) The rescaled metrics $\omega_{CY,t}\in\frac{1}{|\log|t||}c_{1}(\mathcal{O}(1))$ have nontrivial finite diameter Gromov Hausdorff subsequential limits.

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  There is an affine structure on the essential skeleton $Sk(X)$ away from a Hausdorff codimension two singular subset. On the smooth locus, there is a Riemannian metric obtained as the Hessian of local solutions to the real Monge-Ampère equation. This metric agrees with the Gromov-Hausdorff limit, which is conjecturally independent of the subsequence.

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  (Topology) Under the strict Calabi-Yau condition $h^{p,0}(X_{t})=0$ for $p<n$, the essential skeleton is homeomorphic to $S^{n}$.

The heuristic idea is that the essential skeleton should be the base of the hypothetical SYZ fibration, and the SYZ fibration is approximated by logarithm maps, at least in the generic region. We briefly comment on the status of the Kontsevich-Soibelman conjecture (cf. also [53, section 4.5]):

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  The uniform diameter estimate independent of small $t$

  $$C^{-1}\leq\text{diam}(X,\omega_{CY,t})\leq C,$$

  is recently established in joint work with Tosatti [56], using primarily Riemannian geometric methods. This together with Gromov compactness proves the Kähler class normalization prediction.

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  The prediction about the existence of a metrically preferred affine structure away from codimension two on the base, is part of the general lore of the SYZ conjecture (cf. section 2.4), even though there is insufficient evidence. Generally speaking, the simplicial complex structure on $Sk(X)$ induces a piecewise affine structure, and improving it to an affine structure away from codimension two, would require highly nontrivial choices. The author is not aware of a completely satisfactory answer to the following elementary question:

  Question 5.

  Given a one-parameter family of quartic K3 surfaces near the large complex structure limit, without special symmetry, how can we determine the location of singular points on $Sk(X)$? How can we write down the affine structure on the regular locus explicitly?

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  The topology of the essential skeleton is an active research topic in birational geometry. The homeomorphism between $Sk(X)$ with $S^{n}$ is verified for many examples. In general, it is known [62][63] that $Sk(X)$ is a ‘pseudomanifold’, its rational homology groups agree with $S^{n}$, and its fundamental group has trivial profinite completion, but the actual homeomorphism type is still elusive, and supposedly requires appealing to the Poincaré conjecture.