Calabi-Yau metrics with cone singularities along intersecting complex lines: the unstable case

Abstract

We produce local Calabi-Yau metrics on ${\mathbf{C}}^2$ with conical singularities along three or more complex lines through the origin whose cone angles strictly violate the Troyanov condition. The tangent cone at the origin is a flat polyhedral Kähler cone with conical singularities along two intersecting lines: one with cone angle corresponding to the line with smallest cone angle, while the other forms as the collision of the remaining lines into a single conical line. Using a branched covering argument, we can construct Calabi-Yau metrics with cone singularities along cuspidal curves with cone angle in the unstable range.

arXiv commentFinal version. Accepted in the Journal of the London Mathematical Society

References in corpus (4)

• Degenerations of ${\mathbf{C}}^n$ and Calabi-Yau metrics
• Schauder estimates for equations with cone metrics, I
• On the stability of extensions of tangent sheaves on Kähler-Einstein Fano / Calabi-Yau pairs
• Schauder estimates on products of cones