Degeneration of Calabi-Yau metrics and canonical basis

TL;DR

The work develops a cohesive framework for degeneration limits of Calabi–Yau metrics, showing that for polarised CY degenerations with essential skeleton dimension $1\le m\le n$, the $C^0$ hybrid limit of CY potentials coincides with the non-archimedean CY metric on $X_K^{an}$. It introduces a canonical basis of sections under a valuative-independence condition and recasts the limit data as the unique minimiser of a Kontorovich dual functional in an optimal transport problem, thereby linking NA pluripotential theory, Bergman-kernel methods, and OT. The paper also extends the theory to intermediate and large complex structure limits, providing a variational interpretation via a cost function $c(x,p)$ derived from valuations, and discusses the implications for metric SYZ and mirror symmetry through theta-function bases. Collectively, these results offer a principled pathway to understand the metric collapse and limit geometry of CY degenerations, with potential applications to SYZ fibrations and metric mirror duality.

Abstract

For polarised degenerations of Calabi-Yau manifolds whose essential skeleton has dimension $1\leq m\leq n$, we show that the $C^0$ potential theoretic limit of the Calabi-Yau metrics agrees with the non-archimedean Calabi-Yau metric on the Berkovich analytification. Moreover, this limit data can be encoded into the unique minimiser of the Kontorovich functional of an optimal transport problem, under some algebro-geometric assumptions on the existence of a canonical basis of sections for tensor powers of the polarisation line bundle.

Theorems & Definitions (96)

• Conjecture 1.1
• Theorem 1.2
• Remark 1.3
• Definition 1.4
• Remark 1.5
• Remark 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Theorem 2.4