ScalingStacks

Conjecture 1.1 . [03Y3]

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Conjecture 1.1.

[26][16] Let (Xt,gt)(X_{t},g_{t}) be a degenerating family of (polarized) Calabi-Yau 3-folds near the large complex structure limit, equipped with Calabi-Yau metrics gtg_{t}. Then for |t|≪1|t|\ll 1, the 3-fold XtX_{t} admits a special Lagrangian T3T^{3}-fibration over a base BB homeomorphic to a 3-sphere, known as the SYZ fibration. The base BB is equipped with an affine structure and a compatible metric gBg_{B} solving the real Monge-Ampère equation with singularities along a trivalent graph 𝔇⊂B\mathfrak{D}\subset B, such that the rescaled Calabi-Yau metrics (Xt,diam​(gt)−2​gt)(X_{t},\text{diam}(g_{t})^{-2}g_{t}) converge to (B,gB)(B,g_{B}) in Gromov-Hausdorff sense as t→0t\to 0. Morever, suitably away from 𝔇\mathfrak{D} the metrics gtg_{t} are approximated by a semiflat metric up to exponentially small errors.

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