Valuative independence for Calabi--Yau varieties

TL;DR

This paper constructs valuatively independent bases for sections of line bundles on log Calabi--Yau pairs, linking algebraic and tropical geometry through canonical functions on skeletons.

Contribution

It introduces a method to build canonical bases for CY pairs that relate to tropicalizations and degenerations in K-stability.

Findings

Constructed valuatively independent bases for CY pairs.

Bases induce canonical functions on skeletons.

Expected to align with tropicalizations of theta functions.

Abstract

We construct valuatively independent bases for the space of sections of an ample line bundle on a log Calabi--Yau pair over a discretely valued field and the space of regular functions on an affine CY pair with maximal boundary. While the bases are not in general unique, they induce canonical functions on the respective skeletons and are expected to agree with tropicalizations of theta functions when they exist. The proof uses techniques from the study of higher rank degenerations in K-stability.