Due to the analytic difficulty of the SYZ conjecture, especially the nongeneric regions with large Riemannian curvature, the literature has developed many interpretations of the conjectures, with somewhat diverging goals.

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  (Soft versions) For applications to homological mirror symmetry, one is primarily interested in constructing Lagrangian fibrations without requiring $\text{Im}(\Omega)|_{L}=0$, and therefrom build a mirror manifold $X^{\vee}$ and prove the categorical predictions $D^{b}Coh(X^{\vee})\simeq D^{\pi}Fuk(X,\omega)$. This viewpoint separates the symplectic and holomorphic data of the Calabi-Yau manifold, and has a topological/algebraic flavour.

  Remark 1.

  The special Lagrangian condition usually left out of homological mirror symmetry discussions, is supposedly related to Bridgeland stability conditions, which is a popular categorical interpretation of the BPS condition on D-branes.

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  (Strong metric version) On compact Calabi-Yau manifolds near the large complex structure limit, find a global special Lagrangian torus fibration.

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  (Weak metric version) Prove for a suitable class of compact Calabi-Yau manifolds near the large complex structure limit that a special Lagrangian $T^{n}$-fibration exists on a large subset with at least $99\%$ of the Calabi-Yau measure. More precisely, the percentage converges to $100\%$ in the limit.

The metric versions are highly sensitive to the Calabi-Yau metric, which is a much more rigid structure compared to the soft versions. They conform to the PDE spirit of the SYZ paper, while the soft versions are closer to the mirror symmetry motivations of the SYZ conjecture.

The strong metric version is perhaps the most faithful to the original intention of the SYZ paper. Its main evidence comes from the special case of hyperkähler metrics, mentioned in section 2.3. Other peripheral evidence comes from the construction of Lagrangian fibrations in many examples, ignoring the $\text{Im}(\Omega)|_{L}=0$ condition [58]. There are however many subtleties besetting this strong version, mentioned in section 2.4, making the conjecture very formidable, and by comparison the supporting evidence seems inadequate. Notably, the special Lagrangian singularities are not sufficiently understood, and we are not aware of any argument that definitively rules out special Lagrangians intersecting each other in the non-generic regions with large Riemannian curvature. As food for future thought, a somewhat weakened version bypassing these possible objections is

The weak metric version, on the other hand, concerns only the generic region, which is more accessible than the strong version. It conforms to the more cautious expectation, that the special Lagrangian fibration is only a limiting phenomenon. The bulk of the survey will focus on this weak version.