European Journal of Mathematics

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

Examples of tropical-to-Lagrangian correspondence

  • Grigory MikhalkinEmail author
Research Article


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.


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

Mathematics Subject Classification

53D20 53D12 14T05 



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.


  1. 1.
    Cannas da Silva, A.: Lectures on Symplectic Geometry. Lecture Notes in Mathematics, vol. 1764. Springer, Berlin (2001)Google Scholar
  2. 2.
    Cannas de Silva, A.: A Chiang-type lagrangian in \({\mathbb{CP}}^2\). Lett. Math. Phys. 108(3), 699–710 (2018)Google Scholar
  3. 3.
    Caporaso, L.: Algebraic and tropical curves: comparing their moduli spaces. In: Farkas, G., Morrison, I. (eds.) Handbook of Moduli. Vol. I. Advanced Lectures in Mathematics (ALM), vol 24, pp. 119–160. International Press, Somerville (2013)Google Scholar
  4. 4.
    Chiang, R.: New Lagrangian submanifolds of \({\mathbb{CP}}^{n}\). Int. Math. Res. Not. IMRN 2004(45), 2437–2441 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Delzant, T.: Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math. France 116(3), 315–339 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Givental, A.B.: Lagrangian imbeddings of surfaces and the open Whitney umbrella. Funct. Anal. Appl. 20(3), 197–203 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kalinin, N., Mikhalkin, G.: Tropical differential forms (in preparation)Google Scholar
  8. 8.
    Kalinin, N., Shkolnikov, M.: Tropical curves in sandpiles. C. R. Math. Acad. Sci. Paris 354(2), 125–130 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kirby, R.C., Scharlemann, M.G.: Eight faces of the Poincaré homology \(3\)-sphere. Uspekhi Mat. Nauk 37(5(227)), 139–159 (1982) (in Russian)Google Scholar
  10. 10.
    Mak, C.Y., Ruddat, H.: Tropically constructed Lagrangians in mirror quintic threefolds (in preparation)Google Scholar
  11. 11.
    Mandel, T., Ruddat, H.: Descendant log Gromov–Witten invariants for toric varieties and tropical curves (2016). arXiv:1612.02402
  12. 12.
    Mandel, T., Ruddat, H.: Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves (in preparation)Google Scholar
  13. 13.
    Matessi, D.: Lagrangian pairs of pants (2018). arXiv:1802.02993
  14. 14.
    Mikhalkin, G.: Enumerative tropical algebraic geometry in \({\mathbb{R}}^2\). J. Amer. Math. Soc. 18(2), 313–377 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mikhalkin, G.: Tropical geometry and its applications. In: Sanz-Solé, M., et al. (eds.) International Congress of Mathematicians, vol. II, pp. 827–852. European Mathematical Society (EMS), Zürich (2006)Google Scholar
  16. 16.
    Mikhalkin, G.: Quantum indices and refined enumeration of real plane curves. Acta Math. 219(1), 135–180 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nishinou, T.: Toric degenerations, tropical curve, and Gromov–Witten invariants of Fano manifolds. Canad. J. Math. 67(3), 667–695 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nishinou, T., Siebert, B.: Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135(1), 1–51 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Seidel, P.: Graded Lagrangian submanifolds. Bull. Soc. Math. France 128(1), 103–149 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shustin, E.: A tropical approach to enumerative geometry. St. Petersburg Math. J. 17(2), 343–375 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Speyer, D.E.: Parameterizing tropical curves I: curves of genus zero and one. Algebra Number Theory 8(4), 963–998 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tyomkin, I.: Tropical geometry and correspondence theorems via toric stacks. Math. Ann. 353(3), 945–995 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Viro, O.Ya.: Real plane algebraic curves: constructions with controlled topology. Leningrad Math. J 1(5), 1059–1134 (1989)Google Scholar
  24. 24.
    Waldhausen, F.: Eine Klasse von \(3\)-dimensionalen Mannigfaltigkeiten. I. Invent. Math. 3, 308–333 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Waldhausen, F.: Eine Klasse von \(3\)-dimensionalen Mannigfaltigkeiten. II. Invent. Math. 4, 87–117 (1967)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université de Genève, MathématiquesCarougeSwitzerland

Personalised recommendations