Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 

 

Cocycles on tropical varieties via piecewise polynomials


Author: Georges Francois
Journal: Proc. Amer. Math. Soc. 141 (2013), 481-497
MSC (2010): Primary 14T05; Secondary 14C17, 14F99
Published electronically: June 22, 2012
MathSciNet review: 2996952
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We use piecewise polynomials to define tropical cocycles generalising the well-known notion of tropical Cartier divisors to higher codimensions. Groups of cocycles are tropical analogues of Chow cohomology groups. We also introduce an intersection product of cocycles with tropical cycles (the counterpart of the classical cap product) and prove that this gives rise to a Poincaré duality in some cases.


References [Enhancements On Off] (What's this?)

  • [AR1] Lars Allermann and Johannes Rau, First steps in tropical intersection theory, Math. Z. 264 (2010), no. 3, 633–670. MR 2591823, 10.1007/s00209-009-0483-1
  • [AR2] L. Allermann, J. Rau, Tropical rational equivalence on $ \mathbb{R}^r$, arxiv:0811.2860v2.
  • [A1] L. Allermann, Tropical intersection products on smooth varieties, J. Eur. Math. Soc., 74, (2012), no. 7, 707-726.
  • [A2] L. Allermann, Chern classes of tropical vector bundles, to appear in Ark. Mat., arxiv:0911.2909v1.
  • [B] Michel Brion, Piecewise polynomial functions, convex polytopes and enumerative geometry, Parameter spaces (Warsaw, 1994) Banach Center Publ., vol. 36, Polish Acad. Sci., Warsaw, 1996, pp. 25–44. MR 1481477
  • [E] A. Esterov, Tropical varieties with polynomial weights and corner loci of piecewise polynomials, arxiv:1012.5800v3.
  • [F] William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037
  • [FS] William Fulton and Bernd Sturmfels, Intersection theory on toric varieties, Topology 36 (1997), no. 2, 335–353. MR 1415592, 10.1016/0040-9383(96)00016-X
  • [FH] G. Francois, S. Hampe, Universal families of rational tropical curves, arxiv:1105.1674v2, to appear in Canadian Journal of Mathematics.
  • [FR] G. Francois, J. Rau, The diagonal of tropical matroid varieties and cycle intersections, arxiv:1012.3260v1.
  • [GKM] Andreas Gathmann, Michael Kerber, and Hannah Markwig, Tropical fans and the moduli spaces of tropical curves, Compos. Math. 145 (2009), no. 1, 173–195. MR 2480499, 10.1112/S0010437X08003837
  • [K] E. Katz, Tropical intersection theory from toric varieties, Collect. Math., 63 (2012), no. 7, 23-44.
  • [KP] Eric Katz and Sam Payne, Piecewise polynomials, Minkowski weights, and localization on toric varieties, Algebra Number Theory 2 (2008), no. 2, 135–155. MR 2377366, 10.2140/ant.2008.2.135
  • [O] James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR 1207587
  • [P] Sam Payne, Equivariant Chow cohomology of toric varieties, Math. Res. Lett. 13 (2006), no. 1, 29–41. MR 2199564, 10.4310/MRL.2006.v13.n1.a3
  • [R] J. Rau, Intersections on tropical moduli spaces, arxiv:0812.3678v1.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14T05, 14C17, 14F99

Retrieve articles in all journals with MSC (2010): 14T05, 14C17, 14F99


Additional Information

Georges Francois
Affiliation: Fachbereich Mathematik, Technische Universität Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
Email: gfrancois@email.lu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11359-0
Received by editor(s): March 29, 2011
Received by editor(s) in revised form: July 5, 2011
Published electronically: June 22, 2012
Additional Notes: The author is supported by the Fonds National de la Recherche (FNR), Luxembourg.
Communicated by: Lev Borisov
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.