Conifold transitions and mori theory

Abstract

We show there is a symplectic conifold transition of a projective 3-fold which is not deformation equivalent to any Kähler manifold. The key ingredient is Mori's classification of extremal rays on smooth projective 3-folds. It follows that there is a (nullhomologous) Lagrangian sphere in a projective variety which is not the vanishing cycle of any Kähler degeneration, answering a question of Donaldson.