Minimal rational curves on wonderful group compactifications

Brion, Michel; Fu, Baohua

doi:10.5802/jep.20

Minimal rational curves on wonderful group compactifications
[Courbes rationnelles minimales sur les compactifications magnifiques des groupes]

Brion, Michel ¹ ; Fu, Baohua ²

¹ Institut Fourier, Université de Grenoble B.P. 74, 38402 Saint-Martin d’Hères Cedex, France
² Institute of Mathematics, AMSS 55 ZhongGuanCun East Road, Beijing, 100190, China

Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 153-170.

Résumé
Abstract

Soient $G$ un groupe algébrique simple et $X$ sa compactification magnifique. Nous montrons que $X$ possède une unique famille de courbes rationnelles minimales, et nous décrivons explicitement la sous-famille formée des courbes passant par un point général. Nous en déduisons une propriété de rigidité de $X$ , lorsque $G$ n’est pas de type $A_{1}$ ou $C$ .

Consider a simple algebraic group $G$ of adjoint type, and its wonderful compactification $X$ . We show that $X$ admits a unique family of minimal rational curves, and we explicitly describe the subfamily consisting of curves through a general point. As an application, we show that $X$ has the target rigidity property when $G$ is not of type $A_{1}$ or $C$ .

| 1 citation dans Numdam

DOI : 10.5802/jep.20

Classification : 14L30, 14M27, 20G20
Keywords: Minimal rational curves, wonderful compactifications
Mot clés : Courbes rationnelles minimales, compactifications magnifiques

Affiliations des auteurs :

Brion, Michel ¹ ; Fu, Baohua ²

¹ Institut Fourier, Université de Grenoble B.P. 74, 38402 Saint-Martin d’Hères Cedex, France
² Institute of Mathematics, AMSS 55 ZhongGuanCun East Road, Beijing, 100190, China

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Brion, Michel; Fu, Baohua. Minimal rational curves on wonderful group compactifications. Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 153-170. doi : 10.5802/jep.20. http://www.numdam.org/articles/10.5802/jep.20/

Bibliographie
Cité par

[BGMR11] Bravi, P.; Gandini, J.; Maffei, A.; Ruzzi, A. Normality and non-normality of group compactifications in simple projective spaces, Ann. Inst. Fourier (Grenoble), Volume 61 (2011) no. 6, p. 2435-2461 (2012) | DOI | Numdam | MR | Zbl

[BK05] Brion, M.; Kumar, S. Frobenius splitting methods in geometry and representation theory, Progress in Math., 231, Birkhäuser Boston, Inc., Boston, MA, 2005, pp. x+250 | MR | Zbl

[Bou07] Bourbaki, N. Éléments de mathématique. Groupes et algèbres de Lie, Springer, Berlin, 2006–2007, pp. 138 | DOI | Zbl

[Bri07] Brion, M. The total coordinate ring of a wonderful variety, J. Algebra, Volume 313 (2007) no. 1, pp. 61-99 | DOI | MR | Zbl

[CFH14] Chen, Y.; Fu, B.; Hwang, J.-M. Minimal rational curves on complete toric manifolds and applications, Proc. Edinburgh Math. Soc. (2), Volume 57 (2014) no. 1, pp. 111-123 | DOI | MR | Zbl

[CP11] Chaput, P. E.; Perrin, N. On the quantum cohomology of adjoint varieties, Proc. London Math. Soc. (3), Volume 103 (2011) no. 2, pp. 294-330 | DOI | MR | Zbl

[DCP83] De Concini, C.; Procesi, C. Complete symmetric varieties, Invariant theory (Montecatini, 1982) (Lect. Notes in Math.), Volume 996, Springer, Berlin, 1983, pp. 1-44 | DOI | Zbl

[Dem77] Demazure, M. Automorphismes et déformations des variétés de Borel, Invent. Math., Volume 39 (1977) no. 2, pp. 179-186 | MR | Zbl

[FH12] Fu, B.; Hwang, J.-M. Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity, Invent. Math., Volume 189 (2012) no. 2, pp. 457-513 | DOI | MR | Zbl

[HM02] Hwang, Jun-Muk; Mok, Ngaiming Deformation rigidity of the rational homogeneous space associated to a long simple root, Ann. Sci. École Norm. Sup. (4), Volume 35 (2002) no. 2, pp. 173-184 | DOI | Numdam | MR | Zbl

[HM04a] Hwang, Jun-Muk; Mok, Ngaiming Birationality of the tangent map for minimal rational curves, Asian J. Math., Volume 8 (2004) no. 1, pp. 51-63 | DOI | MR | Zbl

[HM04b] Hwang, Jun-Muk; Mok, Ngaiming Deformation rigidity of the $20$ -dimensional $F_{4}$ -homogeneous space associated to a short root, Algebraic transformation groups and algebraic varieties (Encyclopaedia Math. Sci.), Volume 132, Springer, Berlin, 2004, pp. 37-58 | DOI | MR | Zbl

[HM05] Hwang, Jun-Muk; Mok, Ngaiming Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation, Invent. Math., Volume 160 (2005) no. 3, pp. 591-645 | DOI | MR | Zbl

[Hor69] Horrocks, G. Fixed point schemes of additive group actions, Topology, Volume 8 (1969), pp. 233-242 | MR | Zbl

[Hwa01] Hwang, J.-M. Geometry of minimal rational curves on Fano manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000) (ICTP Lect. Notes), Volume 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, pp. 335-393 | MR | Zbl

[Kan99] Kannan, Senthamarai S. Remarks on the wonderful compactification of semisimple algebraic groups, Proc. Indian Acad. Sci. Math. Sci., Volume 109 (1999) no. 3, pp. 241-256 | DOI | MR | Zbl

[Keb02] Kebekus, S. Families of singular rational curves, J. Algebraic Geom., Volume 11 (2002) no. 2, pp. 245-256 | DOI | MR | Zbl

[Kol96] Kollár, J. Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3), 32, Springer-Verlag, Berlin, 1996, pp. viii+320 | DOI | MR | Zbl

[LM03] Landsberg, J. M.; Manivel, L. On the projective geometry of rational homogeneous varieties, Comment. Math. Helv., Volume 78 (2003) no. 1, pp. 65-100 | DOI | MR | Zbl

[Lun73] Luna, D. Slices étales, Sur les groupes algébriques (Mém. Soc. Math. France (N.S.)), Volume 33, Société Mathématique de France, Paris, 1973, pp. 81-105 | Numdam | MR | Zbl

[Tim03] Timashëv, D. A. Equivariant compactifications of reductive groups, Mat. Sb., Volume 194 (2003) no. 4, pp. 119-146 | DOI | MR | Zbl

[Vai84] Vainsencher, I. Complete collineations and blowing up determinantal ideals, Math. Ann., Volume 267 (1984) no. 3, pp. 417-432 | DOI | MR | Zbl

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