Counting Rational Curves and Standard Complex Structures on HyperK\"ahler ALE 4-manifolds

TL;DR

This paper classifies hyperK"ahler ALE 4-manifolds with finite group actions, counts rational curves on them, and shows their twistor spaces are non-K"ahlerian, extending Kronheimer's results with new examples.

Contribution

It specifies complex structures containing rational curves on hyperK"ahler ALE manifolds and proves their twistor spaces cannot be K"ahlerian, providing new non-compact examples.

Findings

Count of rational curves varies lower semi-continuously across parameter space.

Identified points where complex structures include the minimal resolution.

Proved twistor spaces of hyperK"ahler ALE manifolds are non-K"ahlerian.

Abstract

All hyperK\"ahler ALE 4-manifolds with a given non-trivial finite group $\Gamma$ in $SU(2)$ at infinity are parameterized by an open dense subset of a real linear space of dimension $3$rank$\Phi$. Here, $\Phi$ denotes the root system associated with $\Gamma$ 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 hyperK\"ahlerian family of complex structures includes the complex structure of the minimal resolution. Furthermore, we count the rational curves lying on each hyperK\"ahler 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 hyperK\"ahler ALE cannot be K\"ahlerian. In particular, we strengthen some results of Kronheimer (J. Differential Geom., 29(3):665--683, 1989) and provide examples of non-compact and non-K\"ahlerian twistor spaces.