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Amoebas of algebraic varieties and curves counting (following G. Mikhalkin). (Amibes de variétés algébriques et dénombrement de courbes (d’après G. Mikhalkin).) (French) Zbl 1059.14067
Bourbaki seminar. Volume 2002/2003. Exposes 909–923. Paris: Société Mathématique de France (ISBN 2-85629-156-2/pbk). Astérisque 294, 335-361, Exp. No. 921 (2004).
This is a survey of a recent development of the theory of amoebas of algebraic varieties and its applications to real and enumerative algebraic geometry. The amoeba of a complex algebraic subvariety of \(({\mathbb C}^*)^n\) is its image in \({\mathbb R}^n\) by the coordinate-wise valuation map \(z\mapsto\log| z| \). Amoebas were introduced by Gelfand, Kapranov, and Zelevinsky in 1994, and since then a number of interesting geometric properties of amoebas have been discovered. Using the theory of amoebas, G. Mikhalkin [Ann. Math. (2) 151, 309–326 (2000, Zbl 1073.14555)] classified real plane algebraic curves with the maximal number of ovals and maximal intersection with few real lines. In a similar way one can define non-Archimedean amoebas, i.e., amoebas of algebraic varieties defined over a field with a real non-Archimedean valuation. In general, the non-Archimedean amoebas are polyhedral complexes (for example, graphs for algebraic curves), and their geometry (so-called “tropical geometry”) reveals a very deep and interesting relation with the geometry of algebraic varieties. Among various aspects of this theory, the author discussed a striking application to the computation of Gromov-Witten invariants of toric surfaces, which was proposed by Kontsevich and successfully realized by G. Mikhalkin [C. R., Math., Acad. Sci. Paris 336, No. 8, 629–634 (2003; Zbl 1027.14026)]. As result, the count of specific algebraic curves in toric surfaces is reduced to the count of certain lattice paths in the corresponding convex lattice polygons.
For the entire collection see [Zbl 1052.00010].

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14P25 Topology of real algebraic varieties
12J25 Non-Archimedean valued fields
14J26 Rational and ruled surfaces
14N10 Enumerative problems (combinatorial problems) in algebraic geometry