Examples of tropical-to-Lagrangian correspondence

The paper associates Lagrangian submanifolds in symplectic toric varieties to certain tropical curves inside the convex polyhedral domains of 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...

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Bibliographic Details
Published in:European journal of mathematics 2019-09, Vol.5 (3), p.1033-1066
Main Author: Mikhalkin, Grigory
Format: Article
Language:English
Subjects:
Algebraic Geometry
Domains
Homology
Manifolds (mathematics)
Mathematics
Mathematics and Statistics
Research Article
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Summary:The paper associates Lagrangian submanifolds in symplectic toric varieties to certain tropical curves inside the convex polyhedral domains of 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 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 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.
ISSN:2199-675X
2199-6768
DOI:10.1007/s40879-019-00319-6