×

zbMATH — the first resource for mathematics

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].

MSC:
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
PDF BibTeX XML Cite