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.