Small angle limits of negatively curved Kahler-Einstein metrics with crossing edge singularities

TL;DR

This paper studies the behavior of negatively curved Kähler-Einstein metrics with crossing edge singularities on log canonical pairs as some cone angles approach zero, showing smooth convergence away from divisors and Gromov-Hausdorff convergence near divisors.

Contribution

It extends Guenancia's theorem to crossing edge singularities, demonstrating convergence of these metrics to mixed cusp and edge metrics as cone angles tend to zero.

Findings

Metrics converge smoothly away from divisors when cone angles go to zero.

Near divisors, metrics converge in Gromov-Hausdorff sense to mixed cylinder and edge metrics.

The results generalize previous work to crossing edge singularities with varying cone angle limits.

Abstract

Let $(X, D)$ be a log smooth log canonical pair such that $K_X+D$ is ample. Extending a theorem of Guenancia and building on his techniques, we show that negatively curved K\"{a}hler-Einstein crossing edge metrics converge to K\"{a}hler-Einstein mixed cusp and edge metrics smoothly away from the divisor when some of the cone angles converge to $0$. We further show that near the divisor such normalized K\"{a}hler-Einstein crossing edge metrics converge to a mixed cylinder and edge metric in the pointed Gromov-Hausdorff sense when some of the cone angles converge to $0$ at (possibly) different speeds.