SYZ mirror conjecture for Calabi–Yau manifolds

Let $X$ be a Calabi–Yau manifold. A Lagrangian fibration is a map $\pi:X\to B$ onto a topological manifold $B$ whose fibers are graded with respect to a holomorphic volume form $\Omega$. Let $\mathscr Y$ be an analytic space over the Novikov field $\Lambda=\mathbb C((T^{\mathbb R}))$, and let $f:\mathscr Y\to B$ be a tropically continuous map onto the same base. The maps $\pi$ and $f$ have a common singular locus skeleton $\Delta\subset B$, and write $B_0=B\setminus\Delta$, $\pi_0=\pi|_{B_0}$, and $f_0=f|_{B_0}$. SYZ mirror conjecture. Given any Calabi–Yau manifold $X$, there exist such $\pi$ and $f$ satisfying:

• $\pi$ and $f$ have the same singular locus skeleton $\Delta$ in $B$;
• $\pi_0$ and $f_0$ induce the same integral affine structures on $B_0$;
• $f_0$ is isomorphic to the canonical dual affinoid torus fibration $\pi_0^\vee$ associated to $\pi_0$.

This conjecture formulates the expected SYZ construction by requiring a Lagrangian fibration and its non-archimedean mirror fibration to share the same affine base and singularity skeleton. The paper’s main theorem verifies that the explicitly constructed varieties $X$ and $Y$ in the conifold transition setting are SYZ mirror under this definition, while the conjecture is stated for arbitrary Calabi–Yau manifolds.