Donaldson–Scaduto conjecture on collapsing special Lagrangians

Let XtX_t be a 11-parameter family of compact Calabi–Yau manifolds with metrics gtg_t admitting special Lagrangian fibration structures and with bounded diameters, and let CtC_t be a holomorphic curve in XtX_t. Assume that XtX_t are collapsing, meaning

(Xi,gi)GH(B,g),(X_i,g_i)\xrightarrow{GH}(B_{\infty},g_{\infty}),

where BB_{\infty} is an integral affine manifold equipped with a Hessian-type metric outside a codimension-two locus. Donaldson–Scaduto conjecture. The curves CtC_t converge to a tropical curve on BB_{\infty}. This folklore conjecture motivates the use of tropical geometry in collapsing Calabi–Yau manifolds. The paper says it has been proved only in specific settings, including algebraic curves in (C)2(\mathbb{C}^*)^2 with incidence relations and torus fiber bundles, and remains unresolved for compact SU(n)\operatorname{SU}(n)-Calabi–Yau manifolds.

Sources & referencesView supporting material

Primary source

Shih-Kai Chiu and Yu-Shen Lin, “Special Lagrangian submanifolds in K3-fibered Calabi-Yau 3-folds”, arXiv:2410.17662 (2024).

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