Degenerations of Negative Kähler-Einstein Surfaces

Abstract

Every compact Kähler manifold with negative first Chern class admits a unique metric $g$ such that $\text{Ric}(g)=-g$. Understanding how families of these metrics degenerate gives insight into their geometry and is important for understanding the compactification of the moduli space of negative Kähler-Einstein metrics. I study a special class of such families in complex dimension two. Following the work of Sun and Zhang (2019) in the Calabi-Yau case, I construct a Kähler-Einstein neck region interpolating between canonical metrics on components of the central fiber. This provides a model for the limiting geometry of metrics in the family.