Amibes de variétés algébriques et dénombrement de courbes  [ Amoebas of algebraic varieties and curves counting ]
Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 921, p. 335-361

Amoebas of algebraic varieties in ( * ) n are the images of these varieties under the moment map Log :( * ) n n , Log :(z 1 ,...,z n )(log|z 1 |,...,log|z n |). G. Mikhalkin’s results show the usefulness of amoebas in the study of real and complex algebraic varieties. Amoebas can be deformed to certain polyhedral complexes which are called tropical algebraic varieties. This deformation gives a possibility to compute Gromov-Witten invariants of the projective plane and other toric surfaces by counting tropical curves.

Les amibes des variétés algébriques dans ( * ) n sont les images de ces variétés par l’application des moments Log :( * ) n n , Log :(z 1 ,...,z n )(log|z 1 |,...,log|z n |). Des résultats obtenus par G. Mikhalkin montrent l’utilité des amibes pour l’étude des variétés algébriques réelles et complexes. Les amibes peuvent être déformées en des complexes polyédraux appelés variétés algébriques tropicales. Cette déformation permet, en particulier, de calculer les invariants de Gromov-Witten du plan projectif et d'autres surfaces toriques en dénombrant des courbes tropicales.

Classification:  14P25,  14N10,  32A60,  32Q55,  14N35
Keywords: amoebas of algebraic varieties, non-archimedian amoebas, tropical geometry, Gromov-Witten invariants
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     author = {Itenberg, Ilia},
     title = {Amibes de vari\'et\'es alg\'ebriques et d\'enombrement de courbes},
     booktitle = {S\'eminaire Bourbaki : volume 2002/2003, expos\'es 909-923},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {294},
     year = {2004},
     note = {talk:921},
     pages = {335-361},
     zbl = {1059.14067},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2002-2003__45__335_0}
}
Itenberg, Ilia. Amibes de variétés algébriques et dénombrement de courbes, in Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 921, pp. 335-361. http://www.numdam.org/item/SB_2002-2003__45__335_0/

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