The non-archimedean Calabi–Yau metric convergence conjecture

From papers

Let XDtX\to \mathbb{D}^*_t be a polarised meromorphic degeneration of nn-dimensional Calabi–Yau manifolds, with relatively ample line bundle LL, and let XKanX_K^{an} be the associated Berkovich analytification for K=C((t))K=\mathbb{C}((t)). Write Sk(X)Sk(X) for the essential skeleton, of dimension m1m\geq 1, and let hCY,th_{CY,t} and CY,0\lVert\cdot\rVert_{CY,0} denote respectively the fibrewise Hermitian Calabi–Yau metric and the semipositive non-archimedean Calabi–Yau metric on LL. Non-archimedean Calabi–Yau metric convergence conjecture. After suitable normalisation, the norm function hCY,t1logt|\cdot|_{h_{CY,t}}^{\frac{1}{|\log |t||}} on LXtL\to X_t converges in the C0C^0-hybrid topology to CY,0\lVert\cdot\rVert_{CY,0} on LXKanL\to X_K^{an} as t0t\to 0. This conjectural convergence would identify the collapsing complex Calabi–Yau metrics with their non-archimedean pluripotential-theoretic limit; the source presents it as a general expectation, and gives no resolution.

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Sources & referencesView supporting material

Primary source

Yang Li, “Degeneration of Calabi-Yau metrics and canonical basis”, arXiv:2505.11087 (2025).

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