Singularities of plane complex curves and limits of K\"ahler metrics with cone singularities. I: Tangent Cones

TL;DR

This paper classifies tangent cones at points of limit spaces arising from sequences of K"ahler-Einstein metrics with cone singularities in two complex dimensions, enhancing understanding of their geometric limits.

Contribution

It provides a classification of tangent cones for non-collapsed limits of K"ahler-Einstein metrics with cone singularities in complex dimension two.

Findings

Classification of tangent cones achieved

Identification of possible geometric structures at singular points

Insights into the local geometry of limit spaces

Abstract

We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of K\"ahler-Einstein metrics with cone singularities.