The SYZ conjecture for mirror Calabi–Yau manifolds

Let $X$ and $\check{X}$ be Calabi–Yau manifolds which are mirror to each other. A special Lagrangian torus fibration is a fibration with torus fibers calibrated by the special Lagrangian condition; suppose that $\rho:X\to B$ and $\check{\rho}:\check{X}\to B$ are such fibrations with sections. The SYZ conjecture. Both $X$ and $\check{X}$ admit special Lagrangian torus fibrations with sections over the same base, whose regular fibers over the same point $b\in B$ are dual tori, and there exist Fourier-type transforms responsible for interchanging symplectic-geometric and complex-geometric data between $X$ and $\check{X}$. This is the geometric proposal of Strominger, Yau and Zaslow explaining mirror symmetry through T-duality; its general formulation remains a guiding conjectural framework.