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

Abstract

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