Abstract
We introduce the formalism of positive super currents on \({\mathbb{R}^{n}}\), in strong analogy with the theory of positive currents in \({\mathbb{C}^{n}}\). We consider intersection of currents and Lelong numbers, and as an application we show that the formalism can be used to describe tropical varieties. This is similar in spirit to the fact that in complex analysis the current of integration of an analytic variety can be identified with a closed, positive current.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bedford E., Taylor B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)
Berezin F.A.: Introduction to Superanalysis. D. Reidel, Dordrecht (1987)
Berndtsson B.: Subharmonicity properties of the bergman kernel and some other functions associated to pseudoconvex domains. Ann. Inst. Fourier (Grenoble) 56, 1633–1662 (2006)
Demailly, J.P.: Complex Analytic and Differential Geometry. http://www.fourier.ujf-grenoble.fr/demailly/ (2009)
Einsiedler M., Kapranov M., Lind M.: Non-archimedean amoebas and tropical varieties. J. Reine Angew. Math. 601, 139–157 (2006)
Gathmann A.: Tropical algebraic geometry. Jahresber. Deutsch. Math.-Verein. 108(1), 3–32 (2006)
Klartag, B.: Marginals of geometric inequalities. In: Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics (2007)
Klimek, M.: Pluripotential theory. In: London Mathematical Society Monographs. New Series, 6. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1991)
Mikhalkin, G.: Amoebas of algebraic varieties and tropical geometry. In: Different Faces of Geometry, pp. 257–300. Int. Math. Ser. (N.Y.), 3. Kluwer/Plenum, New York (2004)
Mikhalkin, G.: Tropical geometry and its applications. In: International Congress of Mathematicians, vol. II, pp. 827–852. Eur. Math. Soc., Zürich (2006)
Passare M., Rullgård H.: Amoebas, monge-ampère measures, and triangulations of the newton polytope. Duke Math. J. 141, 787–810 (2004)
Rashkovskii A.: Indicators for plurisubharmonic functions of logarithmic growth. Indiana Univ. Math. J. 50(3), 1433–1446 (2001)
Rashkovskii A.: Total masses of mixed monge-ampère currents. Mich. Math. J. 51(1), 169–185 (2003)
Rauch J., Taylor B.A.: The Dirichlet problem for the multidimensional monge-ampère equation. Rocky Mt. J. Math. 7, 345–364 (1977)
Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. In: Idempotent Mathematics and Mathematical Physics, Proceedings in Vienna 2003, Contemp. Math., vol. 377. American Mathematical Society (2005)
Sturmfels, B.: Solving systems of polynomial equations. In: CBMS Regional Conference Series in Mathematics, vol. 97 (2002)