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Product-metric limit on S^1 × S^2 for Calabi–Yau metrics on deformations of a singular complete intersection

Prove that for a one-parameter deformation X_ε of the singular complete intersection Calabi–Yau threefold X^*, the Calabi–Yau metric g_CY^ε in the Fubini–Study Kähler class, restricted to the real locus L ≅ S^1 × S^2, converges in the C^∞ topology to a product metric g_product on S^1 × S^2 as ε → 0.

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Background

Motivated by numerical evidence on the complete intersection Calabi–Yau example CICY2, the authors observe that the approximate harmonic 1-form on its real locus L ≅ S1 × S2 behaves like a product of an almost constant form along the S1-direction and near-zero along the S2-direction. This suggests that as the smoothing parameter ε tends to the singular limit, the metric on L approaches a product structure.

Based on these observations, the authors explicitly formulate a conjecture asserting C∞ convergence of the restricted Calabi–Yau metrics to a product metric on S1 × S2 as ε → 0.

References

We therefore make the following conjecture:

Conjecture Let X_\epsilon be a 1-parameter deformation of the singular variety X* from subsection:intersections-of-quadrics-and-quartics, and denote by g_{\text{CY}\epsilon the uniquely determined Calabi-Yau metric in Kähler class \omega_{\text{FS} \mid {X\epsilon}. Then, we have g_{\text{CY}\epsilon \mid_L \rightarrow g_{\text{product} \text{ in } C\infty \text{ as } \epsilon \rightarrow 0, where g_{\text{product} denotes a product metric on S1 \times S2.

— Harmonic $1$-forms on real loci of Calabi-Yau manifolds (2405.19402 - Douglas et al., 29 May 2024) in Section 5 (Experimental Results), subsection "Approximate harmonic 1-forms", paragraph "CICY2"