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Bubbling limits of non collapsing polarized K3 surfaces

Itsuki Tazoe

TL;DR

The work provides an explicit, period-map–driven classification of bubbling limits for non-collapsing polarized K3 surface degenerations. By introducing two interlinked bubbling trees—one combinatorial in period data and one geometric in limit spaces—the authors establish a poset isomorphism between them, identifying each bubbling limit with a Kronheimer affine ALE gravitational instanton determined by local period data. This yields a complete, algebro-geometric description of bubbling limits, affirming the de Borbon–Spotti conjecture in the K3 case and validating Odaka’s algebro-geometric construction as genuine bubbling limits. The results are reinforced with explicit A_k examples and comparisons to local models, linking deep differential-geometric degeneration phenomena to tractable algebraic data.

Abstract

We give an explicit and complete description of bubbling limits of a non-collapsing limit of polarized K3 surfaces in terms of the period mapping. In particular, we show that bubbling limits only depend on algebro-geometric data of the given family. As a corollary, this gives an affirmative answer to a conjecture of de Borbon--Spotti and confirms that Odaka's algebro-geometric candidate gives genuine bubbling limits in K3 surfaces case.

Bubbling limits of non collapsing polarized K3 surfaces

TL;DR

The work provides an explicit, period-map–driven classification of bubbling limits for non-collapsing polarized K3 surface degenerations. By introducing two interlinked bubbling trees—one combinatorial in period data and one geometric in limit spaces—the authors establish a poset isomorphism between them, identifying each bubbling limit with a Kronheimer affine ALE gravitational instanton determined by local period data. This yields a complete, algebro-geometric description of bubbling limits, affirming the de Borbon–Spotti conjecture in the K3 case and validating Odaka’s algebro-geometric construction as genuine bubbling limits. The results are reinforced with explicit A_k examples and comparisons to local models, linking deep differential-geometric degeneration phenomena to tractable algebraic data.

Abstract

We give an explicit and complete description of bubbling limits of a non-collapsing limit of polarized K3 surfaces in terms of the period mapping. In particular, we show that bubbling limits only depend on algebro-geometric data of the given family. As a corollary, this gives an affirmative answer to a conjecture of de Borbon--Spotti and confirms that Odaka's algebro-geometric candidate gives genuine bubbling limits in K3 surfaces case.

Paper Structure

This paper contains 20 sections, 26 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convergence notions in Riemannian Geometry
  4. Kronheimer's theory on ALE hyperkähler gravitational instantons
  5. Period theory of ALE hyperkähler gravitational instantons
  6. The moduli spaces and its metric completion of Kähler K3 surfaces
  7. Singularities and period mapping of ADE-singularities
  8. Bubbling trees of polarized K3 surfaces
  9. A localization of periods of K3 surfaces to ALE gravitational instantons
  10. Bubbling trees from periods
  11. Bubbling trees from Gromov-Hausdorff limits
  12. Main result
  13. Geometry of the bubbling limits
  14. Proof of the main theorem
  15. Examples
  16. ...and 5 more sections

Key Result

Theorem 1

Let be a flat proper family of polarized K3 surfaces over the unit disc $\Delta \subset \mathbb{C}$. Assume that the central fiber $X_0$ has ADE-type singularities. For each fiber $(X_t, L_t)$, take the Ricci flat Kähler metric $g_t$ with the Kähler form $\omega_t \in c_1(L_t)$ (for the central fiber $X in the sense of Gromov-Hausdorff.

Theorems & Definitions (55)

  • Theorem 1: Theorem 8. KT, see also Proposition 6.7 in OO and Anderson
  • Theorem 2: Theorem 3. dBS
  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 4
  • ...and 45 more