COMMENTS ON CONIFOLDS

Recently it has been shown that there are paths on the moduli space between two Calabi-Yau manifolds with different topology that have finite length. A priori, there exists the possibility that the singular manifold that is common to two different moduli spaces has two different Ricci-flat Kähler metrics, each being the limit of the Ricci-flat Kähler metric of the respective topologically distinct Calabi-Yau manifold. In this paper we show that this is not the case and the topology changing path is continuous even in the space of Ricci-flat Kähler metrics. The explicit form of the Ricci-flat Kähler metrics is calculated in the vicinity of the nodes for the conifold, the resolution and the deformation. A preliminary discussion of global issues is presented and it is shown that, owing to a topological obstruction, the manifold obtained as the result of independently resolving and deforming the nodes of a conifold in general cannot be Kähler. © 1990.