1996/10/04 by Alexander V. Sardo Infirri, Infirri, Alexander V Sardo
Mathematics · Physics and Astronomy · #32M05 (Secondary) #32S45 (Primary) 32L07 #53C07 #Advanced Differential Geometry Research #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.alg-geom/9610004
openalex publication_date 1996/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Γ be a finite group acting linearly on \Cn, freely outside the origin, and let N be the number of conjugacy classes of Γ minus one. A construction of Kronheimer of moduli spaces Xζ of translation-invariant Γ-equivariant instantons on \C2 is generalised to \Cn. The moduli spaces Xζ depend on a parameter ζ∈\QN. The following results are proved: for ζ=0, X0 is isomorphic to \Cn/Γ; if ζ≠ 0, the natural maps Xζ→ X0 are partial resolutions. The moduli Xζ are furthermore shown to admit Kähler metrics which are Asymptotically Locally Euclidean (ALE). A description of the singularities of Xζ using deformation complexes is given, and is applied in particular to the case Γ⊂\SU(3). It is conjectured that for general Γ and generic ζ that the singularities of Xζ are at most quadratic. When Γ⊂\SU(3) a natural holomorphic 3-form is constructed on the smooth locus of Xζ, which is conjectured to be non-vanishing. The morphims Xζ→ X0 are expected to be crepant resolutions and Xζ 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 Xζ with representation moduli of McKay quivers, and describes them completely in the case of abelian groups.