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A smörgåsbord of scalar-flat Kähler ALE surfaces

  • Michael T. Lock EMAIL logo and Jeff A. Viaclovsky


There are many known examples of scalar-flat Kähler ALE surfaces, all of which have group at infinity either cyclic or contained in SU(2). The main result in this paper shows that for any non-cyclic finite subgroup Γ U(2) containing no complex reflections, there exist scalar-flat Kähler ALE metrics on the minimal resolution of 2/Γ, for which Γ occurs as the group at infinity. Furthermore, we show that these metrics admit a holomorphic isometric circle action. It is also shown that there exist scalar-flat Kähler ALE metrics with respect to some small deformations of complex structure of the minimal resolution. Lastly, we show the existence of extremal Kähler metrics admitting holomorphic isometric circle actions in certain Kähler classes on the complex analytic compactifications of the minimal resolutions.

This article is dedicated to the memory of Egbert Brieskorn.

Award Identifier / Grant number: DMS-1405725

Funding statement: The first author was partially supported by NSF Grant DMS-1148490. The second author was partially supported by NSF Grant DMS-1405725.


The authors would like to thank Nobuhiro Honda and Claude LeBrun for many enlightening discussions on scalar-flat Kähler metrics. We would also like to thank Claudio Arezzo, David Calderbank, and Michael Singer for many helpful comments. Also, discussions with Akira Fujiki, Ryushi Goto, and Hans-Joachim Hein on deformations of complex structure were very helpful. Finally, thanks are given to the anonymous referees for numerous suggestions to improve the exposition of the paper.


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Received: 2015-07-10
Revised: 2015-11-26
Published Online: 2016-06-01
Published in Print: 2019-01-01

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