Symplectic conifold transitions
Abstract
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-Kaehler symplectic 6-folds with c_1=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.
Cite
@article{arxiv.math/0209319,
title = {Symplectic conifold transitions},
author = {I. Smith and R. P. Thomas and S. -T. Yau},
journal= {arXiv preprint arXiv:math/0209319},
year = {2007}
}
Comments
Corrections for publication in J. Diff. Geom. and additions to examples. In particular, to date there are no examples proven to be non-Kaehler