Symplectic conifold transitions

Details Symplectic conifold transitions Symplectic conifold transitions

2002-09-24

arxiv.org/abs/math/0209319v2

J.Diff.Geom. 62 (2002) 209-242

Mathematics

Symplectic Geometry

Corrections for publication in J. Diff. Geom. and additions to examples. In particular, to date there are no examples proven t

Scientific paper

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.

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