Special-Lagrangian fibration and corrected dual for the quintic

Fix a one-parameter MUM degeneration $f:\mathcal{X}\to \mathrm{\Delta }$ whose general fibre $X_t$ is a smooth quintic threefold, together with specified Ricci-flat Kähler metrics normalized by ${\int }_{X_t}{\omega }_t^3\mathrm{/}3!=1$. Construct, for sufficiently small $t\mathrm{\ne }0$, a smooth SYZ fibration on a specified open subset $X_t^{\circ }\subseteq X_t$, with special-Lagrangian $T^3$ fibres over the complement of an explicitly controlled discriminant. Here the discriminant is to be a specified closed, locally finite codimension-at-least-two subset of the three-dimensional base with its integral-affine monodromy data; the fibration must be a proper submersion over its complement.

Then form the dual local-system tori $\mathrm{Hom}(H_1(L_b;\mathbb{Z}),U(1))$, specify a completed coefficient ring, wall functions, scattering diagram and theta-function or equivalent correction rule, and prove that the corrected dual compactifies to a complex family isomorphic to the specified mirror quintic family under a stated parameter map. The family, open subset, discriminant, metric normalization, coefficient completion, correction series, and mirror parameter map must all be fixed; no global fibration or correction is implicit in this formulation.

Reviewed as of 2026-11-04. The sources reviewed here do not establish this full target. This is a literature-scope assessment, not a proof that no result exists; the exact target remains the specified metric special-Lagrangian fibration together with the corrected complex-family reconstruction.

The literature contains relevant results at distinct scopes:

• The local Calabi theorem and Ricci-flat corollary supply individual metric existence under their stated compact-Kähler hypotheses. This open problem still takes the normalized metric family as specified data; individual metric existence does not supply the controlled degeneration, special-Lagrangian fibration or corrected mirror reconstruction requested here.
• Yang Li, Theorem 1.1 (arXiv:1912.02360v1), constructs special-Lagrangian torus fibrations on generic regions in the Fermat hypersurface family near the large-complex-structure limit, along a subsequence, with the volume fraction of those regions tending to one. For dimension three this includes a Fermat quintic family. It does not give the specified global complement/discriminant compactification or corrected mirror-family isomorphism.
• Wei-Dong Ruan, Theorem 6.2 (arXiv:math/9909126v2), constructs corresponding Lagrangian torus fibrations for generic quintics near the large-complex limit and mirror quintics near the large-radius limit, with naturally dual regular fibres under the stated symplectic-topological setup. The paper explicitly distinguishes these Lagrangian fibrations from the then-expected special-Lagrangian fibrations (Introduction, pp. 3–5); its theorem therefore does not supply the metric special-Lagrangian family or the correction and compactification data fixed above.
• Yang Li, Theorems 1.4 and 1.8 (arXiv:2605.00516v1), proves a metric-SYZ result for polarized maximal degenerations under the weak comparison property and a valuatively independent basis hypothesis. The reported conclusion gives special-Lagrangian torus fibrations on open subsets of measure at least $1-\delta$ for every $\delta >0$, for sufficiently small parameter. Blum–Liu Theorem 1.1 (arXiv:2604.27890v1) supplies valuatively independent bases for every positive tensor power over a DVR essentially of finite type over a characteristic-zero field, and for sufficiently divisible positive tensor powers over $k\mathrm{⟦}t\mathrm{⟧}$, under its stated projective log-canonical Calabi–Yau and ample-line-bundle hypotheses. Applying this chain to a particular quintic requires checking those exact hypotheses; it is not a proof here.
• Yamamoto Theorem 1.1 (Selecta Math. 32 (2026), arXiv:2404.04972) gives non-Archimedean affinoid torus fibrations outside a codimension-two discriminant for the stated Batyrev–Borisov toric degenerations. That is not a metric special-Lagrangian fibration or a corrected complex mirror reconstruction.
• Gross–Wilson gives K3 and Borcea–Voisin cases at its exact lattice and possibly degenerate-metric scope, not a quintic reconstruction.

Thus the reviewed sources establish partial metric-SYZ results, symplectic-topological Lagrangian duality, and non-Archimedean torus fibrations, each under its own hypotheses. None of the exact statements above supplies all requested metric, discriminant, coefficient-completion, correction, and mirror-isomorphism data simultaneously. That scope comparison is an inference from the cited theorem statements, not a new theorem.

MUM quintic degeneration and normalized Ricci-flat metrics

Log Calabi–Yau pair, toric degeneration and scattering data

Smooth SYZ fibration and dual local-system torus

Ricci-flat metric in every Kähler class when c₁=0