Mathematical Research Letters

Volume 12 (2005)

Number 5

Conifold transitions and Mori theory

Pages: 767 – 778



Alessio Corti (Imperial College London)

Ivan Smith (Cambridge)


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.

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