Abstract

Abstract:

We prove a Kuranishi-type theorem for deformations of complex structures on ALE K\"ahler surfaces. This is used to prove that for any scalar-flat K\"ahler ALE surface, all small deformations of complex structure also admit scalar-flat K\"ahler ALE metrics. A local moduli space of scalar-flat K\"ahler ALE metrics is then constructed, which is shown to be universal up to small diffeomorphisms (that is, diffeomorphisms which are close to the identity in a suitable sense). A formula for the dimension of the local moduli space is proved in the case of a scalar-flat K\"ahler ALE surface which deforms to a minimal resolution of $\Bbb{C}^2/\Gamma$, where $\Gamma$ is a finite subgroup of ${\rm U}(2)$ without complex reflections.

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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 1547-1589
Launched on MUSE
2019-11-02
Open Access
No
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