Einstein metrics and complex singularities
Einstein metrics and complex singularities
复制标题
爱因斯坦度量和复杂奇点
DOI：
10.1007/s00222-003-0344-1
复制
发表时间：
2002
影响因子：
3.1
通讯作者：
M. Singer
中科院分区：
文献类型：
--
作者：
D. Calderbank;M. Singer
本文研究了非紧4-流形上作为复轨道奇点分解的特殊度量的构造。我们的研究在精神上接近于超凯勒引力瞬子的构造，但我们关注的是另一类奇点。我们证明了孤立循环商奇点的任何分解X都有一个完备的标量平坦Kähler度量（它是hyperkähler当且仅当KX是平凡的），并且如果KX是严格nef，则X也有一个完备的（非Kähler的）负标量曲率的自对偶爱因斯坦度量.特别地，在具有任意第二Betti数的单连通非紧4-流形上构造了完备的自对偶Einstein度量，并构造了这些自对偶Einstein度量的变形：它们是由一个真实的变量的自由函数参数化的族，这里构造的所有度量都是环面的（即等距群包含一个2-环面），并且本质上是显式的.关键的建设是显着的事实，复曲面自对偶爱因斯坦度量给出相当普遍的线性偏微分方程的双曲平面。
This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkähler gravitational instantons, but we focus on a different class of singularities. We show that any resolution X of an isolated cyclic quotient singularity admits a complete scalar-flat Kähler metric (which is hyperkähler if and only if KX is trivial), and that if KX is strictly nef, then X also admits a complete (non-Kähler) self-dual Einstein metric of negative scalar curvature. In particular, complete self-dual Einstein metrics are constructed on simply-connected non-compact 4-manifolds with arbitrary second Betti number.Deformations of these self-dual Einstein metrics are also constructed: they come in families parameterized, roughly speaking, by free functions of one real variable.All the metrics constructed here are toric (that is, the isometry group contains a 2-torus) and are essentially explicit. The key to the construction is the remarkable fact that toric self-dual Einstein metrics are given quite generally in terms of linear partial differential equations on the hyperbolic plane.