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It is well known that Kronheimer classified all hyperkähler ALE $4$-manifolds. In particular, any such manifold must be diffeomorphic to the minimal resolution of $\mathbb C^2/\Gamma$ for a finite group $\Gamma \subset SU(2)$.

Given a manifold $M$ which is diffeomorphic to the minimal resolution of $\mathbb C^2/\Gamma$, is the ALE metric unique on $M$, up to isometry and rescaling?

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It is proved by Kronheimer ( https://projecteuclid.org/euclid.jdg/1214443066 ) that the ALE metric is unique up to isometry if you fix the Kähler classes. (It is necessary, else you can give different volumes to the various exceptional curves and you get non-isometric metrics).

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  • $\begingroup$ What do you mean by "give different volumes to the various exceptional curves"? $\endgroup$
    – Totoro
    Commented Mar 8, 2019 at 18:57
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    $\begingroup$ In the minimal resolution of $\mathbb{C}^2/\Gamma$, viewed as a complex manifold, the singular point is replaced by a chain of holomorphic spheres (whose dual graph is the Dynkin diagram of type $\Gamma$). From a complex algebraic point of view, these spheres are complex projective lines, in particular complex curves. Given a metric on this geometry, these spheres will have some (real two-dimensional) volumes. $\endgroup$
    – user25309
    Commented Mar 11, 2019 at 8:15

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