ScalingStacks

Subsection [04X3]

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(2.2) Kontsevich and Soibelman postulated that Sk⁡(X)\mathrm{Sk}(X) should be the base of the non-archimedean SYZ fibration, but the definition of Sk⁡(X)\mathrm{Sk}(X) does not provide us with a map Xan→Sk⁡(X)X^{\mathrm{an}}\to\mathrm{Sk}(X). To construct such a map, we will use an alternative description of the essential skeleton that appeared in [NX16a]. Let 𝒳\mathscr{X} be a minimal dlt-model of XX, and denote by 𝒳snc\mathscr{X}^{\mathrm{snc}} the open subscheme of 𝒳\mathscr{X} consisting of the points where 𝒳\mathscr{X} is regular and 𝒳k\mathscr{X}_{k} has strict normal crossings. Then the dual intersection complex Δ⁡(𝒳snc)\Delta(\mathscr{X}^{\mathrm{snc}}) of 𝒳ksnc\mathscr{X}^{\mathrm{snc}}_{k} can be canonically embedded into XanX^{\mathrm{an}} (see [MN15, §3]). It follows from [NX16a, 3.3.3] that the image of this embedding is exactly the essential skeleton Sk⁡(X)\mathrm{Sk}(X). To be precise, it is assumed in the statement of [NX16a, 3.3.3] that 𝒳\mathscr{X} is ℚ\mathbb{Q}-factorial and defined over an algebraic curve, but these assumptions are not used in the proof. If the minimal dlt-model 𝒳\mathscr{X} is good, we will now construct a continuous retraction ρ𝒳:Xan→Sk⁡(X)\rho_{\mathscr{X}}\colon X^{\mathrm{an}}\to\mathrm{Sk}(X) by generalizing the construction for snc-models in [MN15, 3.1.5].

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