Deformation of LeBrun's ALE metrics with negative mass

TL;DR

This paper studies how LeBrun's scalar-flat Kähler ALE metrics on complex line bundles over CP^1 can be deformed within a one-dimensional family, altering the complex structure while preserving certain geometric properties.

Contribution

It identifies a one-dimensional family of deformations of LeBrun's ALE metrics with non-standard complex structures, expanding understanding of their geometric flexibility.

Findings

The metric admits a one-dimensional deformation family.

Deformations involve changing the complex structure from the standard one.

The scalar-flat Kähler property is preserved under these deformations.

Abstract

In this article we investigate deformations of a scalar-flat K\"ahler metric on the total space of complex line bundles over CP^1 constructed by C. LeBrun. In particular, we find that the metric is included in a one-dimensional family of such metrics on the four-manifold, where the complex structure in the deformation is not the standard one.