Asymptotic torus-bundle model conjecture for maximal Calabi–Yau degenerations

Let $({\cal X}_{mer},vol)=(X_q,vol_q)$ be a one-parameter family of maximally degenerate Calabi–Yau manifolds, let $X_q^{new}$ be the rescaled family, and let $(X^{\varepsilon,t},J_{X^{\varepsilon}})$ be the twisted torus-bundle model over the limiting affine base, with $q=e^{-1/\varepsilon}$. Asymptotic torus-bundle model conjecture. There exist $C>0$ and a function $t(q)$ such that, with $\varepsilon(q)=C(\log|q|)^{-1}$, the Kähler manifolds $X_q^{new}$ and $X^{\varepsilon(q),t(q)}$ satisfy the stated Gromov–Hausdorff convergence and the embeddings $j_q$ identify their scalar products and complex structures up to uniformly $o(1)$ terms on the smooth parts, after deleting a neighborhood of the singular locus. This gives a precise version of the expected leading asymptotic torus-fibration model; the source gives no resolution.