The SYZ conjecture for mirror Calabi–Yau varieties

Let $X$ and $\check{X}$ be mirror Calabi–Yau varieties. An affine manifold with singularities is a manifold $B$ equipped with an integral affine structure away from a singular locus. The maps

$$\phi:X\rightarrow B,\qquad \check{\phi}:\check{X}\rightarrow B$$

are fibrations.

SYZ conjecture. There is an affine manifold with singularities $B$ and maps $\phi:X\rightarrow B$ and $\check{\phi}:\check{X}\rightarrow B$ which are dual special Lagrangian fibrations.

This is a foundational geometric formulation of mirror symmetry, predicting that mirror Calabi–Yau varieties arise from dual torus fibrations over a common affine base. The source presents it as the geometric motivation for the Gross–Siebert program; no resolution status is given here.