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

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 $ζ\in\Q^N$. The following results are proved: for $\zeta =0$, $X_0$ is isomorphic to $\C^n/Γ$; if $\zeta \ne 0$, the natural maps $X_{\zeta }\to X_0$ are partial resolutions. The moduli $X_{\zeta }$ are furthermore shown to admit Kähler 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 $Γ\subset\SU(3)$. It is conjectured that for general $\Gamma$ and generic $\zeta$ that the singularities of $X_{\zeta }$ are at most quadratic. When $Γ\subset\SU(3)$ 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.

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