Partial Resolutions of Orbifold Singularities via Moduli Spaces of HYM-type Bundles

TL;DR

This paper generalizes Kronheimer's construction of moduli spaces of instantons to higher dimensions, providing partial resolutions of orbifold singularities with ALE metrics and conjecturing smoothness and crepant properties for generic parameters.

Contribution

It extends the moduli space construction to ^n, describes their singularities, and explores their geometric properties, including ALE metrics and potential crepant resolutions.

Findings

For ^n/, the moduli spaces are partial resolutions of singularities.

The moduli spaces admit ALE Ka4hler metrics.

Conjectures on smoothness, crepant resolutions, and singularity types for generic parameters.

Abstract

Let $\Gamma$ be a finite group acting linearly on $\C^n$, freely outside the origin, and let $N$ be the number of conjugacy classes of $\Gamma$ minus one. A construction of Kronheimer of moduli spaces $X_\zeta$ of translation-invariant $\Gamma$-equivariant instantons on $\C^2$ is generalised to $\C^n$. The moduli spaces $X_\zeta$ depend on a parameter $\zeta\in\Q^N$. The following results are proved: for $\zeta=0$, $X_0$ is isomorphic to $\C^n/\Gamma$; if $\zeta\neq 0$, the natural maps $X_\zeta\to X_0$ are partial resolutions. The moduli $X_\zeta$ are furthermore shown to admit K\"ahler metrics which are Asymptotically Locally Euclidean (ALE). A description of the singularities of $X_\zeta$ using deformation complexes is given, and is applied in particular to the case $\Gamma\subset\SU(3)$. It is conjectured that for general $\Gamma$ and generic $\zeta$ that the singularities of $X_\zeta$ are at most quadratic. When $\Gamma\subset\SU(3)$ a natural holomorphic 3-form is constructed on the smooth locus of $X_\zeta$, which is conjectured to be non-vanishing. The morphims $X_\zeta\to X_0$ are expected to be crepant resolutions and $X_\zeta$ to be smooth for generic choices of the parameter $\zeta$. Related open problems in higher-dimensional complex geometry are also mentioned. The paper has a companion paper which identifies the moduli $X_\zeta$ with representation moduli of McKay quivers, and describes them completely in the case of abelian groups.