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A continuous cusp closing process for negative Kähler-Einstein metrics

Xin Fu, Hans-Joachim Hein, Xumin Jiang

TL;DR

The paper constructs a continuous cusp-closing for a family of KE metrics on a smoothing of a cubic-type elliptic singularity embedded as degree-$6$ surfaces in $ ext{CP}^3$, revealing a complex hyperbolic cusp in the limit. The main method is a delicate gluing, combining a Tian–Yau gravitational instanton end with a cusp model via a new neck region, and solving a Monge–Ampère equation with a one-dimensional obstruction space. The authors establish precise asymptotics, Ricci-potential decay, and weighted Hölder estimates, proving that the limit metric on the regular part converges while a TY bubble forms at the cusp tip. This cuspidal gluing framework extends the cusp-closing paradigm to higher-dimensional complex geometry and provides a rigorous mechanism for tracking curvature and topology loss through the cusp, with potential generalizations to broader families of singular canonically polarized surfaces.

Abstract

We give an example of a family of smooth complex algebraic surfaces of degree $6$ in $\mathbb{CP}^3$ developing an isolated elliptic singularity. We show via a gluing construction that the unique Kähler-Einstein metrics of Ricci curvature $-1$ on these sextics develop a complex hyperbolic cusp in the limit, and that near the tip of the forming cusp a Tian-Yau gravitational instanton bubbles off.

A continuous cusp closing process for negative Kähler-Einstein metrics

TL;DR

The paper constructs a continuous cusp-closing for a family of KE metrics on a smoothing of a cubic-type elliptic singularity embedded as degree- surfaces in , revealing a complex hyperbolic cusp in the limit. The main method is a delicate gluing, combining a Tian–Yau gravitational instanton end with a cusp model via a new neck region, and solving a Monge–Ampère equation with a one-dimensional obstruction space. The authors establish precise asymptotics, Ricci-potential decay, and weighted Hölder estimates, proving that the limit metric on the regular part converges while a TY bubble forms at the cusp tip. This cuspidal gluing framework extends the cusp-closing paradigm to higher-dimensional complex geometry and provides a rigorous mechanism for tracking curvature and topology loss through the cusp, with potential generalizations to broader families of singular canonically polarized surfaces.

Abstract

We give an example of a family of smooth complex algebraic surfaces of degree in developing an isolated elliptic singularity. We show via a gluing construction that the unique Kähler-Einstein metrics of Ricci curvature on these sextics develop a complex hyperbolic cusp in the limit, and that near the tip of the forming cusp a Tian-Yau gravitational instanton bubbles off.
Paper Structure (41 sections, 43 theorems, 425 equations, 3 figures, 2 tables)

This paper contains 41 sections, 43 theorems, 425 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Statement of the Main Theorem
  3. Possible generalizations
  4. Outline of the proof
  5. Acknowledgments
  6. Building blocks of the gluing
  7. Degenerations of projective surfaces of general type
  8. Singular Kähler-Einstein metrics with hyperbolic cusps
  9. Tian-Yau metrics
  10. A new neck region between a Tian-Yau end and a hyperbolic cusp
  11. Estimates asymptotically close to the cusp (green region)
  12. Estimates on the middle neck
  13. Estimates asymptotically close to the Tian-Yau end (orange region)
  14. The glued approximate Kähler-Einstein metric
  15. Setting up the glued metric and the Monge-Ampère equation
  16. ...and 26 more sections

Key Result

Lemma 2.3

There exist $R>0$ and a smooth function $\nu:\mathbb{C}^{3}\setminus \overline{B}_R\to\mathbb{C}$ such that for all $z\in TY_0 \setminus \overline{B}_R$. Moreover, $\partial^k \nu(z) = O(|z|^{-4-k})$ as $|z|\to\infty$ for all $k \geqslant 0$.

Figures (3)

  • Figure 1: $(\mathcal{X}_\sigma, \omega_{KE,\sigma})$ for $0 < |\sigma| \ll 1$. The Main Theorem describes the red part.
  • Figure 2: Cusp metric $\omega_{cusp}$, horn metric $\omega_T$, gluing regions.
  • Figure 3: The seven regions of $\mathcal{X}_\sigma$. For the middle neck $\mathfrak{R}_4$ see also Figure \ref{['fig:cusp_horn']}.

Theorems & Definitions (107)

  • Remark 2.1
  • Definition 2.2
  • Lemma 2.3: CH
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 97 more