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<p>We give an example of a family of smooth complex algebraic surfaces of degree 6 in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_708_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{CP}^{3}$</EquationSource> </InlineEquation> 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.</p>

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A Continuous Cusp Closing Process for Negative Kähler-Einstein Metrics

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月4日

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  • Xin Fu,
  • Hans-Joachim Hein,
  • Xumin Jiang

摘要

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.