Compactification of the moduli space of principally polarized abelian varieties

Michel Brion

Séminaire Bourbaki (2005-2006)

  • Volume: 48, page 1-32
  • ISSN: 0303-1179

Abstract

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The coarse moduli space of principally polarized abelian varieties may be compactified in several ways, via the Satake compactification or the toroidal compactifications. This raises the problem of finding a “modular” compactification of this moduli space in terms of geometric objects that describe the boundary points. One also seeks a compactification of the Torelli morphism that associates with every nonsingular projective curve its jacobian. The talk presents the solution of these problems by V. Alexeev, based on work of Mumford, Faltings-Chai, Nakamura, and Namikawa.

How to cite

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Brion, Michel. "Compactification de l’espace des modules des variétés abéliennes principalement polarisées." Séminaire Bourbaki 48 (2005-2006): 1-32. <http://eudml.org/doc/252152>.

@article{Brion2005-2006,
abstract = {Les variétés abéliennes principalement polarisées admettent un espace des modules grossier qu’on sait compactifier de plusieurs façons (compactification de Satake, compactifications toroïdales). Cependant, le problème s’est posé de construire une compactification “modulaire”en termes d’objets géométriques qui permettent de décrire les points du bord. On souhaite aussi compactifier l’application de Torelli qui à chaque courbe algébrique, projective et lisse, associe sa jacobienne. L’exposé présente la solution de ces problèmes par V. Alexeev, à la suite de travaux de Mumford, Faltings–Chai, Nakamura et Namikawa.},
author = {Brion, Michel},
journal = {Séminaire Bourbaki},
keywords = {abelian variety; principal polarization; coarse moduli space; semi-abelian variety; toric variety; Delaunay and Voronoi decompositions; compactified jacobians},
language = {fre},
pages = {1-32},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Compactification de l’espace des modules des variétés abéliennes principalement polarisées},
url = {http://eudml.org/doc/252152},
volume = {48},
year = {2005-2006},
}

TY - JOUR
AU - Brion, Michel
TI - Compactification de l’espace des modules des variétés abéliennes principalement polarisées
JO - Séminaire Bourbaki
PY - 2005-2006
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 48
SP - 1
EP - 32
AB - Les variétés abéliennes principalement polarisées admettent un espace des modules grossier qu’on sait compactifier de plusieurs façons (compactification de Satake, compactifications toroïdales). Cependant, le problème s’est posé de construire une compactification “modulaire”en termes d’objets géométriques qui permettent de décrire les points du bord. On souhaite aussi compactifier l’application de Torelli qui à chaque courbe algébrique, projective et lisse, associe sa jacobienne. L’exposé présente la solution de ces problèmes par V. Alexeev, à la suite de travaux de Mumford, Faltings–Chai, Nakamura et Namikawa.
LA - fre
KW - abelian variety; principal polarization; coarse moduli space; semi-abelian variety; toric variety; Delaunay and Voronoi decompositions; compactified jacobians
UR - http://eudml.org/doc/252152
ER -

References

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