A Kummer construction for gravitational instantons
type: Item
instance of: scholarly article
title: A Kummer construction for gravitational instantons
zbMATH Open document ID: 1234.53020
author: Olivier Biquard
author: Vincent Minerbe
published in: Communications in Mathematical Physics
publication date: 2011-12-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1005.5133
review text: 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.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 83D05
zbMATH DE Number: 5988059
zbMATH Keywords: gravitational instanton
zbMATH Keywords: Ricci flat non-compact Kähler manifolds
zbMATH Keywords: Ricci flat orbifolds
zbMATH Keywords: resolution of singularities
zbMATH Keywords: ALE instantons
zbMATH Keywords: K3 surface
reviewed by: Dmitri Alekseevsky
MaRDI profile type: Publication
OpenAlex ID: W3104550413
arXiv ID: 1005.5133
cites work: Q3994593
cites work: Fibrés de higgs et connexions intégrables : Le cas logarithmique (diviseur lisse)
cites work: Gravitational instantons of type \(D_{k}\)
cites work: Singular monopoles and gravitational instantons
cites work: Q4376390
cites work: Dihedral singularities and gravitational instantons
cites work: Hodge cohomology of gravitational instantons
cites work: Q4523708
cites work: Q5486938
cites work: The construction of ALE spaces as hyper-Kähler quotients
cites work: A Kummer-type construction of self-dual 4-manifolds
cites work: [Russian Text Ignored]
cites work: Q4278406
cites work: A mass for ALF manifolds
cites work: Rigidity for Multi-Taub-NUT metrics
cites work: Persson's list of singular fibers for a rational elliptic surface
cites work: Existence of complete Kähler Ricci-flat metrics on crepant resolutions
cites work: Complete Kahler Manifolds with Zero Ricci Curvature. I
cites work: Complete Kähler manifolds with zero Ricci curvature. II
cites work: A new proof of the existence of Kähler-Einstein metrics on K3. I
cites work: Q4766214
DOI: 10.1007/S00220-011-1366-Y
description: scientific article