Abstract
Given a symplectic manifold M we study Lagrangian cobordisms \(V\subset E\) where E is the total space of a Lefschetz fibration having M as generic fiber. We prove a generation result for these cobordisms in the appropriate derived Fukaya category. As a corollary, we analyze the relations among the Lagrangian submanifolds \(L\subset M\) that are induced by these cobordisms. This leads to a unified treatment—and a generalization—of the two types of relations among Lagrangian submanifolds of M that were previously identified in the literature: those associated to Dehn twists that were discovered by Seidel (Topology 42(5):1003–1063, 2003) and the relations induced by cobordisms in trivial symplectic fibrations described in our previous work (J Am Math Soc 26(2):295–340, 2013; Geom Funct Anal 24(6):1731–1830, 2014).
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Acknowledgements
The first author thanks Jean-Yves Welschinger for useful discussions concerning the examples in real algebraic geometry. Part of this work was accomplished during a stay at the Simons Center for Geometry and Physics. We thank the SCGP and its staff for their gracious hospitality. We thank the referees for carefully reading an earlier version of the paper and for remarks that were helpful to improve the exposition.
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The second author was supported by an NSERC Discovery grant, a FQRNT Group Research grant, a Simons Fellowship and an Institute for Advanced Study fellowship grant.
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Biran, P., Cornea, O. Cone-decompositions of Lagrangian cobordisms in Lefschetz fibrations. Sel. Math. New Ser. 23, 2635–2704 (2017). https://doi.org/10.1007/s00029-017-0318-6
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DOI: https://doi.org/10.1007/s00029-017-0318-6