Conifold transitions and Mori theory
arXiv:math/0501043
Abstract
We show there is a symplectic conifold transition of a projective 3-fold which is not deformation equivalent to any Kaehler manifold. The key ingredient is Mori's classification of extremal rays on smooth projective 3-folds. It follows that there is a Lagrangian sphere in a projective variety which is not the vanishing cycle of any Kaehler degeneration, answering a question of Donaldson.
This paper has been withdrawn because of a critical error in Corollary 7 (vanishing of the square of the first Chern class need not be preserved by the surgery). This error invalidates the proofs of both the main results. In particular, Donaldson's question remains open