Candelas–de la Ossa metric degeneration conjecture

The conjecture concerns continuity and geometric identification of Ricci-flat metrics as Calabi–Yau varieties undergo conifold flops and transitions. Jian Song’s 2012 work explicitly states that it proves this conjecture in that setting.

Known results

• Song, 2012: the metric completion of the canonical Ricci-flat metric on the smooth locus of a projective Calabi–Yau variety with ordinary double points is homeomorphic to the variety, proving the conjecture for conifold flops and transitions.
• Song, 2011: Ricci-flat metrics on Calabi–Yau manifolds related by extremal transitions or flops converge in the Gromov–Hausdorff topology to a common compact metric space.
• Later work records the conjecture for projective Calabi–Yau $n$-conifolds with crepant resolutions and smoothings.

October 2026 extension

On October 1, 2026, Riley Guyett and Ryan McGowan’s preprint Monge-Ampère degenerations and conifold contractions in Fujiki class C claimed convergence for $\mathbb{Q}$-Gorenstein normal projective Calabi–Yau varieties with ordinary double points in the threefold case, under specified resolution and Fujiki-class hypotheses. This is a claimed extension, not an independent verification.

Current status (as of October 2026): Song’s conifold flop/transition formulation is claimed solved, and a broader threefold extension is claimed, but the new extension remains unverified.