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Tropical Homology

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu–Paris Rive Gauche, Sorbonne Université, 4 Place Jussieu, 75252, Paris Cedex 5, France

    Ilia Itenberg

  2. Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 rue d’Ulm, 75230, Paris Cedex 5, France

    Ilia Itenberg

  3. Department of Mathematics, Universität Wien, Vienna, Austria

    Ludmil Katzarkov

  4. Department of Mathematics, University of Miami, Miami, FL, USA

    Ludmil Katzarkov

  5. Université de Genève, Mathématiques, Villa Battelle, 1227, Carouge, Switzerland

    Grigory Mikhalkin

  6. Kansas State University, 138 Cardwell Hall, Manhattan, KS, 66506, USA

    Ilia Zharkov

  7. National Research University Higher School of Economics, Russian Federation, Moscow, Russia

    Ludmil Katzarkov

Authors
  1. Ilia Itenberg
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  2. Ludmil Katzarkov
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  3. Grigory Mikhalkin
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  4. Ilia Zharkov
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Corresponding author

Correspondence to Ilia Zharkov.

Additional information

Communicated by Ngaiming Mok.

The research was partially supported by the NSF FRG grants DMS-0854989 and DMS-1265228. L.K. was supported by Simons research grant, NSF DMS-150908, ERC Gemis, DMS-1265230, DMS-1201475, OISE-1242272 PASI and Simons collaborative Grant HMS. Research of G.M. was partially supported by the grant TROPGEO of the European Research Council, and by the grants 140666, 141329, 159240, 159581 and NCCR “SwissMAP” of the Swiss National Science Foundation. L.K. and I.Z. are partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. No. 14.641.31.0001.

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Itenberg, I., Katzarkov, L., Mikhalkin, G. et al. Tropical Homology. Math. Ann. 374, 963–1006 (2019). https://doi.org/10.1007/s00208-018-1685-9

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  • DOI: https://doi.org/10.1007/s00208-018-1685-9

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