Strominger--Yau--Zaslow conjecture

Suppose $M$ and $\hat M$ are mirror Calabi--Yau $3$-folds. The conjecture asserts the existence, under some additional conditions, of a compact topological $3$-manifold $B$ and surjective continuous maps

$$f:M\longrightarrow B,\qquad \hat f:\hat M\longrightarrow B,$$

with fibres $X_b=f^{-1}(b)$ and $\hat X_b=\hat f^{-1}(b)$ for $b\in B$. Strominger--Yau--Zaslow conjecture. There should be a dense open subset $B_0\subset B$ such that, for every $b\in B_0$, $X_b$ and $\hat X_b$ are nonsingular special Lagrangian $3$-tori $T^3$ in $M$ and $\hat M$, respectively, and are in some sense dual; for each $b\in\Delta=B\setminus B_0$, the fibres $X_b$ and $\hat X_b$ are expected to be singular special Lagrangian $3$-folds in $M$ and $\hat M$. This conjecture gives a geometric explanation of mirror symmetry by relating mirror Calabi--Yau $3$-folds through dual special Lagrangian torus fibrations, with singular fibres over the discriminant locus. Its precise additional conditions and the singular-fibre picture remain open.