Mathematische Annalen

, Volume 374, Issue 1–2, pp 963–1006 | Cite as

Tropical Homology

  • Ilia Itenberg
  • Ludmil Katzarkov
  • Grigory Mikhalkin
  • Ilia ZharkovEmail author


Given a tropical variety X and two non-negative integers p and q we define a homology group \(H_{p,q}(X)\) which is a finite-dimensional vector space over \({\mathbb {Q}}\). We show that if X is a smooth tropical variety that can be represented as the tropical limit of a 1-parameter family of complex projective varieties, then \(\dim H_{p,q}(X)\) coincides with the Hodge number \(h^{p,q}\) of a general member of the family.



We are grateful to Sergey Galkin and Luca Migliorini for useful discussions and explanations. The present work started during the fall 2009 semester “Tropical geometry” at MSRI, and we would like to thank the MRSI for hospitality and excellent working conditions.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Ilia Itenberg
    • 1
    • 2
  • Ludmil Katzarkov
    • 3
    • 4
    • 7
  • Grigory Mikhalkin
    • 5
  • Ilia Zharkov
    • 6
    Email author
  1. 1.Institut de Mathématiques de Jussieu–Paris Rive GaucheSorbonne UniversitéParis Cedex 5France
  2. 2.Département de Mathématiques et ApplicationsEcole Normale SupérieureParis Cedex 5France
  3. 3.Department of MathematicsUniversität WienViennaAustria
  4. 4.Department of MathematicsUniversity of MiamiMiamiUSA
  5. 5.Université de Genève, MathématiquesCarougeSwitzerland
  6. 6.Kansas State UniversityManhattanUSA
  7. 7.National Research University Higher School of EconomicsRussian FederationRussia

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