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European Journal of Mathematics

, Volume 5, Issue 3, pp 1033–1066 | Cite as

Examples of tropical-to-Lagrangian correspondence

  • Grigory MikhalkinEmail author
Research Article
  • 17 Downloads

Abstract

The paper associates Lagrangian submanifolds in symplectic toric varieties to certain tropical curves inside the convex polyhedral domains of \(\mathbb {R}^n\) that appear as the images of the moment map of the toric varieties. We pay a particular attention to the case \(n=2\), where we reprove Givental’s theorem (Givental in Funct Anal Appl 20(3):197–203, 1986) on Lagrangian embeddability of non-oriented surfaces to \(\mathbb {C}^2\), as well as to the case \(n=3\), where we see appearance of the graph 3-manifolds studied by Waldhausen (I Invent Math 3:308–333, 1967a, II Invent Math 4:87–117, 1967b) as Lagrangian submanifolds. In particular, rational tropical curves in \(\mathbb {R}^3\) produce 3-dimensional rational homology spheres. The order of their first homology groups is determined by the multiplicity of tropical curves in the corresponding enumerative problems.

Keywords

Tropical Lagrangian Tropical correspondence Tropical curves Lagrangian submanifolds Lational homology spheres Graph-manifolds 

Mathematics Subject Classification

53D20 53D12 14T05 

Notes

Acknowledgements

The author had benefited from many useful discussions with Tobias Ekholm, Yakov Eliashberg, Sergey Galkin, Alexander Givental, Ilia Itenberg, Conan Leung, Diego Matessi, Vivek Shende and Oleg Viro.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université de Genève, MathématiquesCarougeSwitzerland

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