The metric SYZ conjecture for large complex structure limits

Let $\pi:\mathfrak{X}\to\Delta$ be a large complex structure limit family of Calabi–Yau $n$-folds, and equip $X_t$ for $t\neq0$ with the Ricci-flat Kähler metrics $\omega_t\in c_1(\mathfrak{L}|_{X_t})$. Define the unit-diameter rescaled metrics

$$\widetilde{\omega}_t=\operatorname{diam}(X_t,\omega_t)^{-2}\omega_t.$$

Metric SYZ conjecture. As $t\to0$, $(X_t,\widetilde{\omega}_t)$ converges in the Gromov–Hausdorff topology to a compact metric space $(Z,d)$ with an open dense subset $Z_0$ such that $d|_{Z_0}$ is induced by a Riemannian metric $g_0$, the real Hausdorff codimension of $Z\setminus Z_0$ is at least $2$, and $Z_0$ carries an integral affine structure for which, in local affine coordinates,

$$g_0=\nabla^2F,$$

where $F$ is a smooth convex function satisfying

$$\det(\nabla^2F)=c\in\mathbb{R}_{>0}.$$

This conjecture identifies the collapsed metric limit with the affine and Monge–Ampère geometry predicted by SYZ mirror symmetry. It remains open in general.