论文标题
Orbifold通过Hyperkahler商的分辨率：$ d_2 $ alf歧管
Orbifold resolution via hyperkahler quotients: the $D_2$ ALF manifold
论文作者
论文摘要
我们提出了Kronheimer对Hyperkahler歧管的构建的无限范围概括，以解决$ \ Mathbb {R}^4 $的平坦Orbifold商。与[Kro89]一样，这些流形被构造为仿射空间的Hyperkahler商。这导致对\ emph {singular epariakiant instantons}的研究。在本文中，我们研究了奇异的nahm数据，以产生$ d_2 $渐近的本地平面（ALF）歧管的家族，作为平坦的Orbifold $（\ Mathbb {r}^3 \ times s^1）/z_2 $的变形。此外，我们还引入了NAHM数据的稳定性概念，并证明了Donaldson-uhlenbeck-yau型定理，以关联真实和复杂的配方。我们使用这些结果来构建非singular $ d_2 $ alf歧管的家族的规范Ehresmann连接。在复杂的配方中，我们在这些$ d_2 $ alf歧管与相应的$ a_1 $ ale歧管之间表现出明确的关系。我们猜想了$ \ Mathbb {r}^{4-r} {4-r} \ times t^r $的一般Orbifold商的类似结构和结果，$ 2 \ le R \ le 4 $。 CASE $ r = 4 $作为Hyperkahler商产生K3歧管。
We propose an infinite-dimensional generalization of Kronheimer's construction of families of hyperkahler manifolds resolving flat orbifold quotients of $\mathbb{R}^4$. As in [Kro89], these manifolds are constructed as hyperkahler quotients of affine spaces. This leads to a study of \emph{singular equivariant instantons} in various dimensions. In this paper, we study singular equivariant Nahm data to produce the family of $D_2$ asymptotically locally flat (ALF) manifolds as a deformation of the flat orbifold $(\mathbb{R}^3 \times S^1)/Z_2$. We furthermore introduce a notion of stability for Nahm data and prove a Donaldson-Uhlenbeck-Yau type theorem to relate real and complex formulations. We use these results to construct a canonical Ehresmann connection on the family of non-singular $D_2$ ALF manifolds. In the complex formulation, we exhibit explicit relationships between these $D_2$ ALF manifolds and corresponding $A_1$ ALE manifolds. We conjecture analogous constructions and results for general orbifold quotients of $\mathbb{R}^{4-r} \times T^r$ with $2 \le r \le 4$. The case $r = 4$ produces K3 manifolds as hyperkahler quotients.
