Degenerations of LeBrun twistor spaces

TL;DR

This paper studies the limits of LeBrun twistor spaces related to self-dual metrics on connected sums of complex projective planes, revealing degenerations leading to different geometric structures including hyper-Kaehler metrics.

Contribution

It explicitly describes three types of degenerations of LeBrun twistor spaces and their associated metric limits, expanding understanding of their geometric transitions.

Findings

Degeneration to LeBrun metrics on (n-1) CP^2

Degeneration to LeBrun metrics on line bundles over CP^1

Degeneration to hyper-Kaehler metrics on resolved rational double points

Abstract

We investigate various limits of the twistor spaces associated to the self-dual metrics on n CP ^2, the connected sum of the complex projective planes, constructed by C. LeBrun. In particular, we explicitly present the following 3 kinds of degenerations whose limits of the metrics are: (a) LeBrun metrics on (n-1) CP ^2$, (b) (Another) LeBrun metrics on the total space of the line bundle O(-n) over CP ^1 (c) The hyper-Kaehler metrics on the small resolution of rational double points of type A_{n-1}, constructed by Gibbons and Hawking.