Limiting Strominger–Yau–Zaslow conjecture

Let $(\mathcal{X}_{t_i},g_i)$ converge to $(X_{\infty},g_{\infty})$. Let $(B_i,d_i)$ be non-empty for large $i$ and converge to $(B_{\infty},d_{\infty})$, where $B_i$ are compactified moduli spaces of special Lagrangian tori equipped with normalized McLean metrics. Limiting Strominger–Yau–Zaslow conjecture. The spaces $B_{\infty}$ and $X_{\infty}$ are isometric up to scaling. Furthermore, there is a subspace $B_{\infty,0}\subseteq B_{\infty}$ with $\Gamma:=B_{\infty}\setminus B_{\infty,0}$ of Hausdorff codimension $2$ in $B_{\infty}$ such that $B_{\infty,0}$ is a Monge–Ampère manifold, with the Monge–Ampère metric inducing $d_{\infty}$ on $B_{\infty,0}$. This is a limiting formulation of SYZ relating the metric degeneration of the Calabi–Yau fibres to the moduli spaces of special Lagrangian tori; its asserted convergence and Monge–Ampère structure remain open in this generality.