Degeneration of Calabi-Yau metrics and canonical basis

TL;DR

This paper establishes a connection between the limits of Calabi-Yau metrics in degenerations and non-archimedean geometry, linking potential theory, optimal transport, and algebraic geometry.

Contribution

It demonstrates that the $C^0$ potential theoretic limit of Calabi-Yau metrics matches the non-archimedean Calabi-Yau metric and encodes this data via an optimal transport problem under certain algebraic assumptions.

Findings

The $C^0$ limit of Calabi-Yau metrics agrees with the non-archimedean metric.

The limit data can be characterized as the minimizer of the Kontorovich functional.

The results connect complex geometric degenerations with non-archimedean and optimal transport frameworks.

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.