Abstract
We consider Gromov’s homological higher convexity for complements of tropical varieties, establishing it for complements of tropical hypersurfaces and curves, and for nonarchimedean amoebas of varieties that are complete intersections over the field of complex Puiseux series. Based on these results, we conjecture that the complement of a tropical variety has this higher convexity, and prove a weak form of this conjecture for the nonarchimedean amoeba of any variety over the complex Puiseux field. One of our main tools is Jonsson’s limit theorem for tropical varieties.
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References
Bushueva, N.A., Tsikh, A.K.: On amoebas of algebraic sets of higher codimension. Proc. Steklov Inst. Math. 279, 52–63 (2012)
Gel’fand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser Boston Inc., Boston (1994)
Gromov, M.: Sign and geometric meaning of curvature. Rend. Sem. Mat. Fis. Milano 61(1991), 9–123 (1994)
Henriques, A.: An analogue of convexity for complements of amoebas of varieties of higher codimension. Adv. Geom. 4(1), 61–73 (2004)
Jonsson, M.: Degenerations of amoebae and Berkovich spaces. Math. Ann. doi:10.1007/s00208-015-1210-3
Katz, E.: Tropical realization spaces for polyhedral complexes. In: Algebraic and Combinatorial Aspects of Tropical Geometry, Contemporary Mathematics, vol. 589, pp. 235–251. American Mathematical Society, Providence (2013)
Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence (2015)
Mikhalkin, G.: Decomposition into pairs-of-pants for complex algebraic hypersurfaces. Topology 43(5), 1035–1065 (2004)
Mikhalkin, G., Rau, J.: Tropical geometry, 2014, Draft of a book (2014)
Nisse, M., Sottile, F.: Higher convexity of coamoeba complements (2014). arXiv:1405.4900
Nisse, M., Sottile, F.: Non-archimedean coamoebae. In: Tropical and Non-Archimedean Geometry, Contemporary Mathematics, vol. 605, pp. 73–91. American Mathematical Society (2014)
Rullgård, H.: Polynomial amoebas and convexity. Technical report, Stockholm University, 2001, Research Reports in Mathematics, no. 8
Speyer, D., Sturmfels, B.: The tropical Grassmannian. Adv. Geom. 4(3), 389–411 (2004)
Acknowledgments
We thank Mattias Jonsson and Eric Katz for their help in understanding their work.
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Research of Sottile is supported in part by NSF grant DMS-1001615. This is based upon work done at the NIMS, Daejeon, Korea, This work was supported by National Institute for Mathematical Sciences 2014 Thematic Program.
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Nisse, M., Sottile, F. Higher convexity for complements of tropical varieties. Math. Ann. 365, 1–12 (2016). https://doi.org/10.1007/s00208-015-1256-2
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DOI: https://doi.org/10.1007/s00208-015-1256-2