# Hyperkähler ALE $4$-manifolds

It is well known that Kronheimer classified all hyperk?hler 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?

• 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. – user25309 Mar 11 at 8:15