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.

**[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*, 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.

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:
https://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.