Amibes de variétés algébriques et dénombrement de courbes

Itenberg, Ilia

Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 921, pp. 335-361.

Les amibes des variétés algébriques dans ${\left({ℂ}^{*}\right)}^n$ sont les images de ces variétés par l’application des moments $\mathrm{Log}:{\left({ℂ}^{*}\right)}^n\to {ℝ}^n$, $\mathrm{Log}:(z_1,...,z_n)\mapsto (log|z_1|,...,log|z_n\left|\right)$. 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.

Amoebas of algebraic varieties in ${\left({ℂ}^{*}\right)}^n$ are the images of these varieties under the moment map $\mathrm{Log}:{\left({ℂ}^{*}\right)}^n\to {ℝ}^n$, $\mathrm{Log}:(z_1,...,z_n)\mapsto (log|z_1|,...,log|z_n\left|\right)$. 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.

Classification : 14P25, 14N10, 32A60, 32Q55, 14N35
Mot clés : amibes de variétés algébriques, amibes non archimédiennes, géométrie tropicale, invariants de Gromov-Witten
Keywords: amoebas of algebraic varieties, non-archimedian amoebas, tropical geometry, Gromov-Witten invariants

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