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Apr 19, 2010 at 12:37 comment added Dmitri Panov Don, I think, that if you want a more positive answer to the question, it is really worth to think what should be the best analogue of a Lesfshetz fibrations for projective manifolds with $\mathbb C^*$ action. If you will remove the restictions on Lefshetz fibration to be gereric (i.e. all singularities are double points), there could be more examples. If a "resonable" definition exists, who knows, maybe it could be generalised to symplectic cathegory to give a postivie answer, following the discussion of Tim
Apr 19, 2010 at 8:36 comment added Don Stanley Thanks for your response. I guess it's possible to come up with examples using toric manifolds and various projections of polytopes onto intervals, but this is not something I really know about.
Apr 16, 2010 at 23:40 history answered Dmitri Panov CC BY-SA 2.5