Complexifying Lie Group Actions on Homogeneous Manifolds of Non-compact Dimension Two

Ahmadi, S. Ruhallah ; Gilligan, Bruce

Canadian mathematical bulletin, Tome 57 (2014) no. 4, pp. 673-682

Résumé

If $X$ is a connected complex manifold with ${{d}_{X}}\,=\,2$ that admits a (connected) Lie group $G$ acting transitively as a group of holomorphic transformations, then the action extends to an action of the complexification $\widehat{G}$ of $G$ on $X$ except when either the unit disk in the complex plane or a strictly pseudoconcave homogeneous complex manifold is the base or fiber of some homogeneous fibration of $X$ .

Détail

DOI : 10.4153/CMB-2014-028-6

Mots-clés : 32M10, homogeneous complex manifold, non-compact dimension two, complexification

Bibliographie

[Abe76] [Abe76] Abels, H., Proper transformation groups. In: Transformation groups. (Proc. Conf., Univ. Newcastle upon Tyne, Newcastle upon Tyne, 1976), London Math. Soc. Lecture Note Series, 26, Cambridge University Press, Cambridge, 1977, pp. 237–248. Google Scholar

[Abe82] [Abe82] Abels, H.,Some topological aspects of proper group actions; noncompact dimension of groups. J. London Math. Soc. (2) 25 (1982), no. 3, 525–538. Google Scholar | DOI

[Akh77] [Akh77] Ahiezer, D. N., Dense orbits with two endpoints. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 308–324, 477; Math. USSR-Izv. 11 (1977), no. 2, 293–307. Google Scholar

[Akh79] [Akh79] Ahiezer, D. N., Algebraic groups that are transitive in the complement to a homogeneous hypersurface. (Russian) Dokl. Akad. Nauk SSSR 245 (1979), no. 2, 281–284; Soviet Math. Dokl. 20 (1979), no. 2, 278–281. Google Scholar

[Akh83] [Akh83] Ahiezer, D. N., Complex n-dimensional homogeneous spaces homotopically equivalent to (2n-2)-dimensional compact manifolds. Selecta Math. Soviet 3(1983/84), no. 3, 286–290. Google Scholar

[Akh13] [Akh13] Ahiezer, D. N., Real group orbits on flag manifolds. In: Lie groups: structure, actions, and representations Progress in Mathematics, 306, Birkhäuser Basel, 2013, pp. 1–24. Google Scholar

[AG94] [AG94] Akhiezer, D. and Gilligan, B., On complex homogeneous spaces with top homology in codimension two.Canad. J. Math. 46 (1994), no. 5, 897–919. Google Scholar | DOI

[BoMo47] [BoMo47] Bochner, S. and Montgomery, D., Groups on analytic manifolds. Ann. of Math. 48 (1947), 659–669. Google Scholar | DOI

[Bor53] [Bor53] Borel, A., Les bouts des espaces homogánes de groupes de Lie. Ann. of Math. (2) 58 (1953), 443–457. Google Scholar | DOI

[Che51] [Che51] Chevalley, C., Théorie des groupes de Lie. Tome II. Groupes algébriques. Actualités Sci. Ind., 1152, Hermann &Cie., Paris, 1951. Google Scholar

[FHW] [FHW] Fels, G., Huckleberry, A. T., and J. A.Wolf, Cycle spaces of flag domains. A complex geometric viewpoint. Progress in Mathematics, 245, Birkhäuser Boston, Inc., Boston, MA, 2006. Google Scholar

[Gil91] [Gil91] Gilligan, B., On the ends of complex manifolds homogeneous under a Lie group. In: Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., 52, American Mathematical Society, Providence, RI, 1991, pp. 217–224. Google Scholar

[Gil95] [Gil95] Gilligan, B., Complex homogeneous spaces of real groups with top homology in codimension two.Ann. Global Anal. Geom. 13 (1995), no. 3, 303–314. Google Scholar | DOI

[GH98] [GH98] Gilligan, B. and Heinzner, P., Globalization of holomorphic actions on principal bundles.Math. Nachr. 189 (1998), 145–156. Google Scholar | DOI

[GH09] [GH09] Gilligan, B. and Huckleberry, A. T., Fibrations and globalizations of compact homogeneous CR-Manifolds.(Russian) Izv. Ross. Akad. Nauk Ser. Mat. 73 (2009), no. 3, 67–126; translation in: Izv. Math. 73 (2009), no. 3, 501–553. Google Scholar | DOI

[Hir70] [Hir70] Hironaka, H., Desingularizations of complex-analytic varieties. (French) In: Actes du Congrás International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 1971, pp. 627–631. Google Scholar

[Hoch65] [Hoch65] Hochschild, G., The structure of Lie groups. Holden-Day Inc., San Franscisco-London-Amsterdam, 1965. Google Scholar

[HO81] [HO81] Huckleberry, A. T. and Oeljeklaus, E., Homogeneous spaces from a complex analytic viewpoint. In: Manifolds and Lie groups. (Notre Dame, Ind., 1980) Progress in Math., 14, Birkhäuser, Boston, MA, 1981, pp. 159–186. Google Scholar

[HO84] [HO84] Huckleberry, A. T. and Oeljeklaus, E., Classification theorems for almost homogeneous spaces. Institut E´ lie Cartan, 9, Universite´ de Nancy, Institut E´ lie Cartan, Nancy, 1984. Google Scholar

[HS81] [HS81] Huckleberry, A. T. and Oeljeklaus, E. Huckleberry, A. T. and Snow, D., A classification of strictly pseudoconcave homogeneous manifolds.Ann. Scuola Norm. Sup. Pisa 8 (1981), no. 2, 231–255. Google Scholar

[HS82] [HS82] Huckleberry, A. T. and Oeljeklaus, E. Huckleberry, A. T. and Snow, D., Almost-homogeneous Kähler manifolds with hypersurface orbits.Osaka J. Math. 19 (1982), no. 4, 763–786. Google Scholar

[Ka67] [Ka67] Kaup, W., Reelle Transformationsgruppen und invariante Metriken auf komplexen Räumen.Invent. Math. 3 (1967), 43–70. Google Scholar | DOI

[Mal75] [Mal75] Malyshev, F. M., Complex homogeneous spaces of semisimple Lie groups of the first category. (Russian) Izv. Akad. Nauk SSSR, Ser. Mat. 39 (1975),no. 5, 992–1002, 1219; translation in: Math. USSR Izv. 9 (1977), no. 5, 939–949. Google Scholar

[Mal77] [Mal77] Malyshev, F. M., Complex homogeneous spaces of semisimple Lie groups of type Dn. (Russian) Izv. Akad. Nauk SSSR, Ser. Mat. 41 (1977), no. 4, 829–852, 959; translation in: Math. USSR Izv. 11 (1977), no. 4, 783–805. Google Scholar

[Mat79] [Mat79] Matsuki, T., The orbits of affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. Japan 31 (1979), no. 2, 331–357. Google Scholar | DOI

[Mat82] [Mat82] Matsuki, T., Orbits on affine symmetric spaces under the action of parabolic subgroups. Hiroshima Math. J. 12 (1982), no. 2, 307–320. Google Scholar

[Mat60] [Mat60] Matsushima, Y., Espaces homogánes de Stein des groupes de Lie complexes.Nagoya Math. J. 16 (1960), 205–218. Google Scholar

[MN63] [MN63] Morimoto, A. and Nagano, T.,On pseudo-conformal transformations of hypersurfaces.J. Math. Soc. Japan. 15 (1963), 289–300. Google Scholar | DOI

[Mos55] [Mos55] Mostow, G. D., On covariant fiberings of Klein spaces. Amer. J. Math. 77 (1955), 247–278. Google Scholar | DOI

[Mos62] [Mos62] Mostow, G. D., Covariant fiberings of Klein spaces. II. Amer. J. Math. 84 (1962), 466–474. Google Scholar | DOI

[OR84] [OR84] Oeljeklaus, K. andRichthofer, W., Homogeneous complex surfaces. Math. Ann. 268 (1984), no. 3, 273–292. Google Scholar | DOI

[Oni60] [Oni60] Onishchik, A. L., Complex hulls of compact homogeneous spaces. (Russian) Dokl. Akad. Nauk SSSR 130 (1960), 726–729; translation in: Soviet Math. Dokl. 1 (1960) 88–91. Google Scholar

[Oni62] [Oni62] Onishchik, A. L., Inclusion relations among transitive compact transformation groups. (Russian) Trudy Mosk. Mat. Obsc. 11 (1962), 199–242; translation in: Amer. Math. Soc. Transl. 50 (1966), 5–58. Google Scholar

[Oni69] [Oni69] Onishchik, A. L., Decompositions of reductive Lie groups. (Russian) Mat. Sb. 80 (122) (1969), 553–599; translation in: Math. USSR Sb. 9 (1969), 515–554. Google Scholar

[Ste82] [Ste82] Steinsiek, M., Transformation groups on homogeneous-rational manifolds. Math. Ann. 260 (1982), no. 4, 423–435. Google Scholar | DOI

[Wan52] [Wan52] Wang, H.-C, Two-point homogeneous spaces.Ann. of Math. (2) 55 (1952), 177–191. Google Scholar | DOI

[Weis66] [Weis66] Weisfeiler, B., On one class of unipotent subgroups of semisimple algebraic groups. arxiv:math/0005149v1; translated from Russian: Uspehi Mat. Nauk 21 (1966), 222–223. Google Scholar

[Wol69] [Wol69] Wolf, J. A., The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components. Bull. Amer. Math. Soc. 75 (1969), 1121–1237. Google Scholar | DOI