Author:
Lock Michael T.,Viaclovsky Jeff A.
Abstract
Abstract
There are many known examples of scalar-flat
Kähler ALE surfaces, all of which have
group at infinity either cyclic or contained in
{{\rm{SU}}(2)}
.
The main result in this paper shows that for any non-cyclic finite subgroup
Γ
\subset
U(2) containing no complex reflections, there exist
scalar-flat Kähler ALE metrics on the minimal
resolution of
\mathbb{C}^{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.
Funder
National Science Foundation
Subject
Applied Mathematics,General Mathematics
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