Amoebas of algebraic varieties and curves counting

Ilia Itenberg

Séminaire Bourbaki (2002-2003)

  • Volume: 45, page 335-362
  • ISSN: 0303-1179

Abstract

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Amoebas of algebraic varieties in ( ℂ * ) n are the images of these varieties under the moment map Log : ( ℂ * ) n → ℝ 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 calledtropical algebraic varieties. This deformation gives a possibility to compute Gromov-Witten invariants of the projective plane and other toric surfaces by counting tropical curves.

How to cite

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Itenberg, Ilia. "Amibes de variétés algébriques et dénombrement de courbes." Séminaire Bourbaki 45 (2002-2003): 335-362. <http://eudml.org/doc/252140>.

@article{Itenberg2002-2003,
abstract = {Les amibesdes variétés algébriques dans $(\mathbb \{C\}^*)^n$ sont les images de ces variétés par l’application des moments $\mathrm \{Log\}:(\mathbb \{C\}^*)^n \rightarrow \mathbb \{R\}^n$, $\mathrm \{Log\}:(z_1, \ldots , z_n) \mapsto (\log |z_1|, \ldots , \log |z_n|)$. 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 appelésvariétés 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.},
author = {Itenberg, Ilia},
journal = {Séminaire Bourbaki},
keywords = {amoebas of algebraic varieties; non-archimedian amoebas; tropical geometry; Gromov-Witten invariants},
language = {fre},
pages = {335-362},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Amibes de variétés algébriques et dénombrement de courbes},
url = {http://eudml.org/doc/252140},
volume = {45},
year = {2002-2003},
}

TY - JOUR
AU - Itenberg, Ilia
TI - Amibes de variétés algébriques et dénombrement de courbes
JO - Séminaire Bourbaki
PY - 2002-2003
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 45
SP - 335
EP - 362
AB - Les amibesdes variétés algébriques dans $(\mathbb {C}^*)^n$ sont les images de ces variétés par l’application des moments $\mathrm {Log}:(\mathbb {C}^*)^n \rightarrow \mathbb {R}^n$, $\mathrm {Log}:(z_1, \ldots , z_n) \mapsto (\log |z_1|, \ldots , \log |z_n|)$. 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 appelésvariétés 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.
LA - fre
KW - amoebas of algebraic varieties; non-archimedian amoebas; tropical geometry; Gromov-Witten invariants
UR - http://eudml.org/doc/252140
ER -

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