Cocycles on tropical varieties via piecewise polynomials
HTML articles powered by AMS MathViewer
- by Georges Francois PDF
- Proc. Amer. Math. Soc. 141 (2013), 481-497 Request permission
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
- Lars Allermann and Johannes Rau, First steps in tropical intersection theory, Math. Z. 264 (2010), no. 3, 633–670. MR 2591823, DOI 10.1007/s00209-009-0483-1
- L. Allermann, J. Rau, Tropical rational equivalence on $\mathbb {R}^r$, arxiv:0811.2860v2.
- L. Allermann, Tropical intersection products on smooth varieties, J. Eur. Math. Soc., 74, (2012), no. 7, 707–726.
- L. Allermann, Chern classes of tropical vector bundles, to appear in Ark. Mat., arxiv:0911.2909v1.
- Michel Brion, Piecewise polynomial functions, convex polytopes and enumerative geometry, Parameter spaces (Warsaw, 1994) Banach Center Publ., vol. 36, Polish Acad. Sci. Inst. Math., Warsaw, 1996, pp. 25–44. MR 1481477
- A. Esterov, Tropical varieties with polynomial weights and corner loci of piecewise polynomials, arxiv:1012.5800v3.
- 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, DOI 10.1515/9781400882526
- William Fulton and Bernd Sturmfels, Intersection theory on toric varieties, Topology 36 (1997), no. 2, 335–353. MR 1415592, DOI 10.1016/0040-9383(96)00016-X
- G. Francois, S. Hampe, Universal families of rational tropical curves, arxiv:1105.1674v2, to appear in Canadian Journal of Mathematics.
- G. Francois, J. Rau, The diagonal of tropical matroid varieties and cycle intersections, arxiv:1012.3260v1.
- 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, DOI 10.1112/S0010437X08003837
- E. Katz, Tropical intersection theory from toric varieties, Collect. Math., 63 (2012), no. 7, 23–44.
- 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, DOI 10.2140/ant.2008.2.135
- James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR 1207587
- Sam Payne, Equivariant Chow cohomology of toric varieties, Math. Res. Lett. 13 (2006), no. 1, 29–41. MR 2199564, DOI 10.4310/MRL.2006.v13.n1.a3
- J. Rau, Intersections on tropical moduli spaces, arxiv:0812.3678v1.
Additional Information
- Georges Francois
- Affiliation: Fachbereich Mathematik, Technische Universität Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
- Email: gfrancois@email.lu
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 481-497
- MSC (2010): Primary 14T05; Secondary 14C17, 14F99
- DOI: https://doi.org/10.1090/S0002-9939-2012-11359-0
- MathSciNet review: 2996952