Compactification de l'espace des modules des variétés abéliennes principalement polarisées  [ Compactification of the moduli space of principally polarized abelian varieties ]
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque no. 311  (2007), Talk no. 952, p. 1-32

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.

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.

Classification:  11H55,  14D22,  14H10,  14H40,  14K10,  14M25
Keywords: abelian variety, principal polarization, coarse moduli space, semi-abelian variety, toric variety, Delaunay and Voronoi decompositions, compactified jacobians
@incollection{SB_2005-2006__48__1_0,
     author = {Brion, Michel},
     title = {Compactification de l'espace des modules des vari\'et\'es ab\'eliennes principalement polaris\'ees},
     booktitle = {S\'eminaire Bourbaki : volume 2005/2006, expos\'es 952-966},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {311},
     year = {2007},
     note = {talk:952},
     pages = {1-32},
     zbl = {1197.14048},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2005-2006__48__1_0}
}
Brion, Michel. Compactification de l'espace des modules des variétés abéliennes principalement polarisées, in Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Talk no. 952, pp. 1-32. http://www.numdam.org/item/SB_2005-2006__48__1_0/

[1] V. Alexeev - “Log canonical singularities and complete moduli of stable pairs”, arXiv : alg-geom/9608013.

[2] -, “On extra components in the functorial compactification of A g , in Moduli of abelian varieties (Texel Island, 1999), Progr. Math., vol. 195, Birkhäuser, Basel, 2001, p. 1-9. | MR 1827015 | Zbl 1058.14061

[3] -, “Complete moduli in the presence of semiabelian group action”, Ann. of Math. (2) 155 (2002), no. 3, p. 611-708. | MR 1923963 | Zbl 1052.14017

[4] -, “Compactified Jacobians and Torelli map”, Publ. Res. Inst. Math. Sci. 40 (2004), no. 4, p. 1241-1265. | MR 2105707 | Zbl 1079.14019

[5] V. Alexeev & I. Nakamura - “On Mumford's construction of degenerating abelian varieties”, Tohoku Math. J. (2) 51 (1999), no. 3, p. 399-420. | MR 1707764 | Zbl 0989.14003

[6] A. Ash, D. Mumford, M. Rapoport & Y. Tai - Smooth compactification of locally symmetric varieties, Math. Sci. Press, Brookline, Mass., 1975, Lie Groups : History, Frontiers and Applications, vol. IV. | MR 457437 | Zbl 0334.14007

[7] S. Bosch, W. Lütkebohmert & M. Raynaud - Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 21, Springer-Verlag, Berlin, 1990. | Article | Zbl 0705.14001

[8] G. Faltings & C.-L. Chai - Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 22, Springer-Verlag, Berlin, 1990, avec un appendice de D. Mumford. | Article | MR 1083353 | Zbl 0744.14031

[9] R. Hartshorne - Algebraic geometry, Graduate Texts in Math., vol. 52, Springer-Verlag, New York, 1977. | Article | MR 463157 | Zbl 0367.14001

[10] S. Keel & S. Mori - “Quotients by groupoids”, Ann. of Math. (2) 145 (1997), no. 1, p. 193-213. | MR 1432041 | Zbl 0881.14018

[11] J. Kollár & N. I. Shepherd-Barron - “Threefolds and deformations of surface singularities”, Invent. Math. 91 (1988), no. 2, p. 299-338. | MR 922803 | Zbl 0642.14008

[12] G. Laumon & L. Moret-Bailly - Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 39, Springer-Verlag, Berlin, 2000. | Article | MR 1771927 | Zbl 0945.14005

[13] D. Mumford - Abelian varieties, Oxford University Press, Oxford, 1970. | MR 282985 | Zbl 0198.25801 | Zbl 0583.14015

[14] -, “An analytic construction of degenerating abelian varieties over complete rings”, Compositio Math. 24 (1972), p. 239-272. | Numdam | MR 352106 | Zbl 0241.14020

[15] D. Mumford, J. Fogarty & F. Kirwan - Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34, Springer-Verlag, Berlin, 1994, 3e édition. | Article | MR 1304906 | Zbl 0797.14004

[16] I. Nakamura - “On moduli of stable quasi abelian varieties”, Nagoya Math. J. 58 (1975), p. 149-214. | Article | MR 393049 | Zbl 0295.14018

[17] -, “Stability of degenerate abelian varieties”, Invent. Math. 136 (1999), no. 3, p. 659-715. | MR 1695209 | Zbl 0982.14026

[18] Y. Namikawa - “A new compactification of the Siegel space and degeneration of Abelian varieties I& II”, Math. Ann. 221 (1976), no. 2, p. 97-141, pp. 201-241. | MR 480537 | Zbl 0327.14013

[19] -, Toroidal compactification of Siegel spaces, Lect. Notes in Math., vol. 812, Springer, Berlin, 1980. | MR 584625 | Zbl 0466.14011

[20] T. Oda & C. S. Seshadri - “Compactifications of the generalized Jacobian variety”, Trans. Amer. Math. Soc. 253 (1979), p. 1-90. | Article | MR 536936 | Zbl 0418.14019

[21] M. C. Olsson - “Canonical compactifications of moduli spaces for abelian varieties”, http://www.ma.utexas.edu/~molsson/.

[22] M. Raynaud - “Spécialisation du foncteur de Picard”, Publ. Math. Inst. Hautes Études Sci. (1970), no. 38, p. 27-76. | Numdam | MR 282993 | Zbl 0207.51602

[23] -, “Compactification du module des courbes”, in Séminaire Bourbaki (1970/1971), Lect. Notes in Math., vol. 244, Springer, Berlin, 1971, exp. no. 385, p. 47-61. | Numdam | MR 437542 | Zbl 0247.14007

[24] C. Traverso - “Seminormality and Picard group”, Ann. Scuola Norm. Sup. Pisa (3) 24 (1970), p. 585-595. | Numdam | MR 277542 | Zbl 0205.50501