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Amoebas of algebraic varieties and tropical geometry. (English) Zbl 1072.14013
Donaldson, Simon (ed.) et al., Different faces of geometry. New York, NY: Kluwer Academic/Plenum Publishers (ISBN 0-306-48657-1/hbk). International Mathematical Series (New York) 3, 257-300 (2004).
The survey presents an account of the theory of amoebas of algebraic varieties, and their applications to complex analysis, geometry of real and complex algebraic varieties, as well as to the enumerative geometry, notably, the computation of Gromov-Witten invariants. The amoeba of an algebraic variety $V$ in the torus $({\Bbb K}^*)^n$, over a field ${\Bbb K}$ with a real valuation, is its image (or its closure) in ${\Bbb R}^n$ under the coordinate-wise valuation map. The first part of the survey is devoted to the amoebas of complex algebraic varieties under the logarithmic moment map $z\mapsto\log\vert z\vert$. The issues of the asymptotic behavior, convexity, relations to the complex analysis are addressed. As application the author demonstrates his result on the geometry of real plane algebraic curves located in a special way with respect to the coordinate lines [Ann. Math. (2) 151, 309--326 (2000; Zbl 1073.14555)]. The second part of the survey addresses the theory of “tropical varieties”, amoebas of algebraic varieties defined over a field with a real non-Archimedean valuation. Tropical varieties can be viewed as algebraic varieties over the semifield of real numbers equipped with the operations of maximum and addition (“tropical semifield”). Geometrically, they are polyhedral complexes of pure dimension equal to the dimension of the underlying algebraic variety, for example, tropical curves are certain piece-wise linear graphs in ${\Bbb R}^n$. The author discusses various aspects of this rapidly developing research area: presentation of tropical varieties as limits of amoebas of complex varieties, relations between the geometry of tropical varieties and properties of algebraic varieties (such as Riemann-Roch theorem for tropical curves). Finally, the author considers the basics of the enumerative tropical geometry and its breakthrough application to counting complex and real algebraic curves in toric surfaces along his recent work [C. R. Math. Acad. Sci. Paris 336, No. 8, 629--634 (2003; Zbl 1027.14026)]. For the entire collection see [Zbl 1051.53001].

##### MSC:
 14D05 Structure of families 14T05 Tropical geometry 12J25 Non-Archimedean valued fields 14M25 Toric varieties, Newton polyhedra 14N35 Gromov-Witten invariants, quantum cohomology, etc. 14P25 Topology of real algebraic varieties 12K10 Semi-fields
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