Open Access
October, 2002 Symplectic Conifold Transitions
I. Smith, R.P. Thomas, S.-T. Yau
J. Differential Geom. 62(2): 209-242 (October, 2002). DOI: 10.4310/jdg/1090950192


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.


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I. Smith. R.P. Thomas. S.-T. Yau. "Symplectic Conifold Transitions." J. Differential Geom. 62 (2) 209 - 242, October, 2002.


Published: October, 2002
First available in Project Euclid: 27 July 2004

zbMATH: 1071.53541
MathSciNet: MR1988503
Digital Object Identifier: 10.4310/jdg/1090950192

Rights: Copyright © 2002 Lehigh University

Vol.62 • No. 2 • October, 2002
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