Limits of conical K\"ahler-Einstein metrics on rank one horosymmetric spaces

TL;DR

This paper studies the behavior of conical K"ahler-Einstein metrics on rank one horosymmetric Fano manifolds as cone angles decrease, revealing convergence patterns to known Einstein and Ricci-flat metrics.

Contribution

It establishes the limiting behavior of conical K"ahler-Einstein metrics on horosymmetric spaces, connecting them to classical Einstein and Ricci-flat metrics.

Findings

Metrics converge to K"ahler-Einstein metrics on the base space.

Rescaled metrics on fibers converge to Stenzel's Ricci-flat metrics.

Limit cone angle remains positive, ensuring convergence.

Abstract

We consider families of conical K\"ahler-Einstein metrics on rank one horosymmetric Fano manifolds, with decreasing cone angles along a codimension one orbit. At the limit angle, which is positive, we show that the metrics, restricted to the complement of that orbit, converge to (the pull-back of) the K\"ahler-Einstein metric on the basis of the horosymmetric homogeneous space, which is a projective homogeneous space. Then we show that, on the symmetric space fibers, the rescaled metrics converge to Stenzel's Ricci flat K\"ahler metrics.