Abstract
AbstractA natural approach to the construction of nearly $$G_2$$
G
2
manifolds lies in resolving nearly $$G_2$$
G
2
spaces with isolated conical singularities by gluing in asymptotically conical $$G_2$$
G
2
manifolds modelled on the same cone. If such a resolution exits, one expects there to be a family of nearly $$G_2$$
G
2
manifolds, whose endpoint is the original nearly $$G_2$$
G
2
conifold and whose parameter is the scale of the glued in asymptotically conical $$G_2$$
G
2
manifold. We show that in many cases such a curve does not exist. The non-existence result is based on a topological result for asymptotically conical $$G_2$$
G
2
manifolds: if the rate of the metric is below $$-3$$
-
3
, then the $$G_2$$
G
2
4-form is exact if and only if the manifold is Euclidean $$\mathbb R^7$$
R
7
. A similar construction is possible in the nearly Kähler case, which we investigate in the same manner with similar results. In this case, the non-existence results is based on a topological result for asymptotically conical Calabi–Yau 6-manifolds: if the rate of the metric is below $$-3$$
-
3
, then the square of the Kähler form and the complex volume form can only be simultaneously exact, if the manifold is Euclidean $$\mathbb R^6$$
R
6
.
Funder
Gottfried Wilhelm Leibniz Universität Hannover
Publisher
Springer Science and Business Media LLC
Reference32 articles.
1. Bär, C.: Real killing spinors and holonomy. Commun. Math. Phys. 154, 509–521 (1993)
2. Bilal, A., Metzger, S.: Compact weak $$G_2$$-manifolds with conical singularities. Nucl. Phys. B 663(1–2), 343–364 (2003)
3. Biquard, O.: Désingularisation de métriques d’Einstein. I. Invent. Math. 192(1), 197–252 (2013)
4. Biquard, O.: Désingularisation de métriques d’Einstein. II. Invent. Math. 204(2), 473–504 (2016)
5. Biquard, O., Hein, H.-J.: The renormalized volume of a 4-dimensional Ricci-flat ALE space, J. Differ. Geom. to appear. arXiv:1901.03647