A continuous cusp closing process for negative Kähler-Einstein metrics

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.

67 pages, 3 figures