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

TL;DR

This paper constructs local Calabi-Yau metrics with cone singularities along intersecting complex lines in ^2, especially in the unstable cone angle range, using branched coverings and polyhedral cone models.

Contribution

It introduces methods to produce Calabi-Yau metrics with cone singularities violating the Troyanov condition, including along cuspidal curves, expanding the understanding of singular Calabi-Yau geometries.

Findings

Constructed local Calabi-Yau metrics with cone singularities along intersecting lines.

Demonstrated metrics with cone angles in the unstable range.

Used branched covering techniques to handle cuspidal singularities.

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 K\"ahler 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.