Abstract
We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relation to rational equivalence of 0-cycles on the mirror rigid analytic space.
Funding statement: Nick Sheridan was partially supported by a Royal Society University Research Fellowship. Ivan Smith was partially supported by a Fellowship from EPSRC and by the National Science Foundation under Grant No. DMS-1440140, whilst in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 semester.
Acknowledgements
Theorem 1.3 realizes a suggestion of Paul Seidel, who also suggested considering cylindrical rather than planar cobordism. We are grateful to Mohammed Abouzaid, Denis Auroux, Jeff Hicks, Daniel Huybrechts, Joe Rabinoff, Dhruv Ranganathan, Tony Scholl, Paul Seidel, Harold Williams and Tony Yue Yu for helpful conversations.
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