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Algebraic geometry of bubbling Kahler metrics

Yuji Odaka

TL;DR

This work develops an algebro-geometric, non-archimedean framework to model bubbling transitions in Kahler metrics with Euclidean volume growth during degenerations to log terminal singularities. It proves a two-step birational construction that yields algebraic minimal bubbles, $\mathcal{X}'_{min,0}$ and $\mathcal{X}''_{min,0}$, as affine log terminal cones deforming from a fixed central fiber, with normalized volumes strictly increasing along the process; these bubbles relate to analytic bubbles in Song–Dai-Sun contexts via K-stability and cone degenerations. The paper provides coordinate- and valuation-based realizations of minimal bubbles, establishes partial identifications with differential-geometric bubbling in key cases (log curves, ADE/A_k singularities, and 1D degenerations), and extends the framework to negative-weight deformations and base changes (Novikov-type) to achieve canonical forms. It also outlines generalizations to deeper bubbles for klt degenerations and to semi-log canonical degenerations via galaxy models, suggesting a rich non-archimedean picture for moduli and Calabi–Yau metrics. Overall, the results deepen the bridge between algebraic degenerations, K-stability, and metric bubbling, with implications for K-moduli and the algebraic understanding of geometric limits.

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].

Algebraic geometry of bubbling Kahler metrics

TL;DR

This work develops an algebro-geometric, non-archimedean framework to model bubbling transitions in Kahler metrics with Euclidean volume growth during degenerations to log terminal singularities. It proves a two-step birational construction that yields algebraic minimal bubbles, and , as affine log terminal cones deforming from a fixed central fiber, with normalized volumes strictly increasing along the process; these bubbles relate to analytic bubbles in Song–Dai-Sun contexts via K-stability and cone degenerations. The paper provides coordinate- and valuation-based realizations of minimal bubbles, establishes partial identifications with differential-geometric bubbling in key cases (log curves, ADE/A_k singularities, and 1D degenerations), and extends the framework to negative-weight deformations and base changes (Novikov-type) to achieve canonical forms. It also outlines generalizations to deeper bubbles for klt degenerations and to semi-log canonical degenerations via galaxy models, suggesting a rich non-archimedean picture for moduli and Calabi–Yau metrics. Overall, the results deepen the bridge between algebraic degenerations, K-stability, and metric bubbling, with implications for K-moduli and the algebraic understanding of geometric limits.

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].
Paper Structure (14 sections, 17 theorems, 25 equations)

This paper contains 14 sections, 17 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Minimal bubble
  3. Reviewing definitions
  4. Construction via deformation theory
  5. Setup and algebraic construction
  6. Statements on the comparison with differential geometry
  7. Details of the algebraic algorithmic construction
  8. Comparison with the differential geometric construction
  9. Negative weight deformation as scaling down
  10. Construction of minimal bubble via coordinates
  11. Valuative approach
  12. Discussions on more general bubbles
  13. Deeper bubbles for klt degenerations
  14. Towards bubbling for semi-log canonical degeneration

Key Result

Theorem 1.1

Consider an arbitrary flat degeneration of pointed $n$-dimensional log terminal (hence normal) singularities $\pi\colon \mathcal{X}(\supset \sigma(\Delta))\to \Delta\ni 0$ (see Setup Setup), which is strictly degenerating in a certain sense (see Theorem bblcst:def). Then, the following hold. We refer to these obtained (non-canonical) total spaces of the new degenerations to $\mathcal{X}'_{\rm min

Theorems & Definitions (45)

  • Theorem 1.1
  • Example 1.2: cf., BKNBando, dBS and Theorem \ref{['AGDG.cl']}
  • Definition 2.1: Song
  • Definition 2.2: ELS
  • Theorem 2.3: Scale up conification cf., DSIILiLiuBlumHSLXLWXLLXXZ2Od24a
  • Theorem 2.4: Algebro-geometric construction of (minimal) bubbles
  • Definition 2.5: Restriction to one parameter families
  • Theorem 2.6: Algebro-geometric construction of (minimal) bubbles - ver2: with canonical base changes
  • Remark 2.7: Locality and logarithmic generalization
  • Theorem 2.8
  • ...and 35 more