Topology of Asymptotically Conical Calabi–Yau and G₂ Manifolds and Desingularization of Nearly Kähler and Nearly G₂ Conifolds

A natural approach to the construction of nearly G₂ manifolds lies in resolving nearly G₂ spaces with isolated conical singularities by gluing in asymptotically conical G₂ manifolds modelled on the same cone. If such a resolution exits, one expects there to be a family of nearly G₂ manifolds, whose endpoint is the original nearly G₂ conifold and whose parameter is the scale of the glued in asymptotically conical G₂ 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 G₂ manifolds: if the rate of the metric is below - 3 , then the G₂ 4-form is exact if and only if the manifold is Euclidean R⁷. 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 R⁶.