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 $\zeta\in\Q^N$. 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.