Topology of asymptotically conical Calabi-Yau and \(G_2\) manifolds and desingularization of nearly Kähler and nearly \(G_2\) conifolds. (English) Zbl 1512.53049
Summary: A natural approach to the construction of nearly \(G_2\) manifolds lies in resolving nearly \(G_2\) spaces with isolated conical singularities by gluing in asymptotically conical \(G_2\) manifolds modelled on the same cone. If such a resolution exits, one expects there to be a family of nearly \(G_2\) manifolds, whose endpoint is the original nearly \(G_2\) conifold and whose parameter is the scale of the glued in asymptotically conical \(G_2\) 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 \(G_2\) manifolds: if the rate of the metric is below \(-3\), then the \(G_2 \, 4\)-form is exact if and only if the manifold is Euclidean \(\mathbb{R}^7\). 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 \(-3\), then the square of the Kähler form and the complex volume form can only be simultaneously exact, if the manifold is Euclidean \(\mathbb{R}^6\).
MSC:
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53C10 | \(G\)-structures |
| 32Q25 | Calabi-Yau theory (complex-analytic aspects) |
| 53C29 | Issues of holonomy in differential geometry |
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