The metric SYZ conjecture for maximally degenerate Calabi–Yau families

Let $\mathcal{X}=\{\mathcal{X}_t\}_t$ be a maximally degenerate family of polarized Calabi–Yau manifolds of dimension $d$ over the punctured disc

$$\mathbb{D}^\ast:=\{t\in\mathbb{C}^\times\mid |t|<1\}.$$

A special Lagrangian $\mathbb{T}^d$-fibration is a fibration by special Lagrangian $d$-tori, and the normalized Calabi–Yau volume refers to the volume normalized as in the metric SYZ setting.

Metric SYZ conjecture. For any $0<\varepsilon\ll1$, there exists $\delta>0$ such that if $|t|<\delta$, then there exists a special Lagrangian $\mathbb{T}^d$-fibration on an open subset of the fiber $\mathcal{X}_t$ whose normalized Calabi–Yau volume is greater than or equal to $1-\varepsilon$.

This asserts that, after sufficiently large degeneration, special Lagrangian torus fibrations occupy arbitrarily close to the full normalized Calabi–Yau volume. The paper proves the metric SYZ conjecture for the toric degenerations it studies, assuming the existence of the relevant solution; Li further reduced it to the NA MA–real MA comparison property.