Algebraic geometry of bubbling Kahler metrics

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].

arXiv commentv4: Mainly expository revision of the first half of v3 (figures added, clarification in a simpler setup e.g., under Gorenstein assumption). The second half is being revised and expanded as a separate sequel