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

Abstract

A 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 .