Algebraic geometry of bubbling Kahler metrics

TL;DR

This paper develops an algebraic and non-archimedean framework to analyze bubbling phenomena in Kahler metrics with Euclidean volume growth, connecting degenerations, birational modifications, and stability theory.

Contribution

It introduces a new algebraic approach to study bubbling in Kahler geometry, linking degenerations with birational modifications and stability theory.

Findings

Constructed birational modifications for degenerating families

Compared algebraic and analytic bubbling constructions

Connected stability theory with bubbling phenomena

Abstract

We give an algebro-geometric or non-archimedean framework to study bubbling phenomena of Kahler metrics with Euclidean volume growth, after [DS17, Sun23, dBS23]. In particular, for any degenerating family to log terminal singularity, we algebraically construct a finite sequence of birational modifications of the family with milder degenerations, and compare with analytic bubbling constructions in loc.cit. We also provide approaches in terms of coordinates and valuations. Our discussion partially depends on the general framework of stability theory in our [Od24b] (arXiv:2406.02489) after [HL14, AHLH23].