Journal of Differential Geometry

Symplectic Conifold Transitions

I. Smith, R.P. Thomas, and S.-T. Yau


We introduce a symplectic surgery in six dimensions which collapses Lagrangian three-spheres and replaces them by symplectic two-spheres. Under mirror symmetry it corresponds to an operation on complex 3-folds studied by Clemens, Friedman and Tian. We describe several examples which show that there are either many more Calabi-Yau manifolds (e.g., rigid ones) than previously thought or there exist "symplectic Calabi-Yaus" — non-Kähler symplectic 6-folds with c1 = 0. The analogous surgery in four dimensions, with a generalisation to ADE-trees of Lagrangians, implies that the canonical class of a minimal complex surface contains symplectic forms if and only if it has positive square.

Article information

J. Differential Geom., Volume 62, Number 2 (2002), 209-242.

First available in Project Euclid: 27 July 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Smith, I.; Thomas, R.P.; Yau, S.-T. Symplectic Conifold Transitions. J. Differential Geom. 62 (2002), no. 2, 209--242. doi:10.4310/jdg/1090950192.

Export citation