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Gromov–Hausdorff limit of maximally unipotent Calabi–Yau degenerations

Establish that for a maximally unipotent degeneration of simply-connected Calabi–Yau manifolds with full SU(n) holonomy, the sequence of fibers equipped with Ricci-flat metrics normalized to have fixed diameter has a convergent subsequence whose Gromov–Hausdorff limit is a metric space homeomorphic to the n-sphere S^n.

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Background

In Appendix A, the authors discuss the differential-geometric perspective on degenerations of Calabi–Yau manifolds via Gromov–Hausdorff limits, relating asymptotic geometry to the dual graph of the central fiber in a semi-stable degeneration.

Motivated by examples and by prior work on large complex structure limits, they state a conjecture attributed to Gross–Wilson and Kontsevich–Soibelman concerning the limiting metric space for maximally unipotent degenerations, i.e., large complex structure points (type IV limits).

Proving this conjecture would clarify the precise metric collapse behavior of Calabi–Yau families at large complex structure and strengthen the geometric underpinnings of mirror symmetry and the SYZ picture in higher dimensions.

References

Conjecture: Let \mathcal{V}\to \mathbf{D} be a maximally unipotent degeneration of simply-connected Calabi--Yau manifolds with full SU(n) holonomy, z_i\in\mathbf{D} with z_i\to 0, and let g_i be a Ricci-flat metric on V_{zi} normalized to have fixed diameter. Then a convergent subsequence of (V_{zi}, g_i) converges to a metric space V_\infty, where V_\infty is homeomorphic to the n-sphere Sn.

— Physics and Geometry of Complex Structure Limits in Type IIB Calabi-Yau Compactifications (2509.07056 - Monnee et al., 8 Sep 2025) in Appendix A: The Dual Graph and Gromov–Hausdorff Limits (Conjecture)