A Kummer construction for gravitational instantons
The authors give a construction of a class of complete Ricci flat Kähler manifolds with one end which has the asymptotics of a torus bundle over \(\mathbb{R}^m,\, m \leq 4\), minus a ball. This class is a generalization of the class of asymptotically locally Euclidean Ricci flat Kähler manifolds. The construction is inspired by the classical Kummer construction forthe \(K3\) surface, which is obtained from the orbifold \(X= \mathbb{T}^4/\mathbb{Z}_2\) by resolving its 16 singularities. The starting point is a Ricci flat orbifold \(X\) which is a finite quotient of some known Ricci flat Kähler manifolds (e.g., \(X = \mathbb{R}^3 \times \mathbb{S}^1/\mathbb{Z}_2, \, \mathbb{R}^2 \times \mathbb{T}^2/\mathbb{Z}_k,\, \mathbb{R}\times \mathbb{T}^3/\mathbb{Z}_2\) ). Then, using some deformation argument, they prove the existence of a complete Ricci-flat Kähler metric on the minimal resolution \(\hat X\) of \(X\) with asymptotics as described above.
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