Arkiv för Matematik

  • Ark. Mat.
  • Volume 50, Number 2 (2012), 237-258.

Chern classes of tropical vector bundles

Lars Allermann

Full-text: Open access

Abstract

We introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Most of the definitions presented here for tropical vector bundles will be contained in Torchiani, C., Line Bundles on Tropical Varieties, Diploma thesis, Technische Universität Kaiserslautern, Kaiserslautern, 2010, for the case of line bundles. Afterwards we use the bounded rational sections of a tropical vector bundle to define the Chern classes of this bundle and prove some basic properties of Chern classes. Finally we give a complete classification of all vector bundles on an elliptic curve up to isomorphisms.

Note

I would like to thank my advisor Andreas Gathmann for numerous helpful discussions.

Article information

Source
Ark. Mat., Volume 50, Number 2 (2012), 237-258.

Dates
Received: 26 April 2010
Revised: 1 October 2010
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907175

Digital Object Identifier
doi:10.1007/s11512-011-0147-3

Mathematical Reviews number (MathSciNet)
MR2961320

Zentralblatt MATH identifier
1273.14129

Rights
2011 © Institut Mittag-Leffler

Citation

Allermann, Lars. Chern classes of tropical vector bundles. Ark. Mat. 50 (2012), no. 2, 237--258. doi:10.1007/s11512-011-0147-3. https://projecteuclid.org/euclid.afm/1485907175


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References

  • Allermann, L., Tropical intersection products on smooth varieties, Preprint, 2009.
  • Allermann, L. and Rau, J., First steps in tropical intersection theory, Math. Z. 264 (2010), 633–670.
  • Allermann, L. and Rau, J., Tropical rational equivalence, Preprint, 2008.
  • Atiyah, M. F., Vector bundles over an elliptic curve, Proc. Lond. Math. Soc. 37 (1957), 414–452.
  • Gathmann, A., Kerber, M. and Markwig, H., Tropical fans and the moduli spaces of tropical curves, Compositio Math. 145 (2009), 173–195.
  • Torchiani, C., Line Bundles on Tropical Varieties, Diploma thesis, Technische Universität Kaiserslautern, Kaiserslautern, 2010.