2018/02/19 by Mikhalkin, Grigory · 1 citation
#FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1802.06473
The paper associates Lagrangian submanifolds in symplectic toric varieties to certain tropical curves inside the convex polyhedral domains of \Rn 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 on Lagrangian embeddability of non-oriented surfaces to \C2, as well as to the case n=3, where we see appearance of the graph 3-manifolds studied by Waldhausen as Lagrangian submanifolds. In particular, rational tropical curves in \R3 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.