Counting Rational Curves and Standard Complex Structures on HyperKähler ALE 4-manifolds

Abstract

All hyperKähler ALE 4-manifolds with a given non-trivial finite group $Γ$ in $SU(2)$ at infinity are parameterized by an open dense subset of a real linear space of dimension $3$rank$Φ$. Here, $Φ$ denotes the root system associated with $Γ$ via the McKay correspondence. Such manifolds are diffeomorphic to the minimal resolution of a Kleinian singularity. By using the period map of the twistor space, we specify those points in the parameter space at which the hyperKählerian family of complex structures includes the complex structure of the minimal resolution. Furthermore, we count the rational curves lying on each hyperKähler ALE 4-manifold. For each point in the parameter space, we can assign an integer equals to the number of complex structures which contains rational curves. We show this integer function on the parameter space is lower semi-continuous. In the end, based on known results, we prove that the twistor space of any hyperKähler ALE cannot be Kählerian. In particular, we strengthen some results of Kronheimer (J. Differential Geom., 29(3):665--683, 1989) and provide examples of non-compact and non-Kählerian twistor spaces.