The Strominger–Yau–Zaslow conjecture for mirror Calabi–Yau manifolds

Let $X$ and $Y$ be “mirror” Calabi-Yau manifolds. There should exist a base manifold $B$ and surjective continuous maps

$$\pi:X\to B,\qquad f:Y\to B,$$

with fibers $L_q=\pi^{-1}(q)$ and $F_q=f^{-1}(q)$, and a closed set $\Delta\subset B$ such that $B\setminus\Delta$ is dense. Strominger–Yau–Zaslow conjecture. Under some additional conditions, for each $b\in B\setminus\Delta$, the fibers $L_q$ and $F_q$ are nonsingular special Lagrangian tori in $X$ and $Y$ that are, in some sense, dual to one another, while for each $b\in\Delta$ they are singular Lagrangian submanifolds. This is a mathematical formulation of mirror symmetry through dual special Lagrangian torus fibrations; the statement leaves the additional conditions and the precise meaning of duality to be established in each setting.