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

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.