Fibrations and globalizations of compact homogeneous CR-manifolds

B. Gilligan ; A. T. Huckleberry

Izvestiya. Mathematics , Tome 73 (2009) no. 3, pp. 501-553.

Résumé

Fibration methods which were previously used for complex homogeneous spaces and CR-homogeneous spaces of special types [1]–[4] are developed in a general framework. These include the $\mathfrak g$-anticanonical fibration in the CR-setting, which reduces certain considerations to the compact projective algebraic case, where a Borel–Remmert type splitting theorem is proved. This leads to a reduction to spaces homogeneous under actions of compact Lie groups. General globalization theorems are proved which enable one to regard a homogeneous CR-manifold as an orbit of a real Lie group in a complex homogeneous space of a complex Lie group. In the special case of CR-codimension at most two, precise classification results are proved and are applied to show that in most cases there exists such a globalization.

Détail

Keywords: complex homogeneous spaces, homogeneous CR-spaces, homogeneous bundles, globalization.

Bibliographie

[1] A. T. Huckleberry, E. Oeljeklaus, Classification theorems for almost homogeneous spaces, Inst. Élie Cartan, 9, Université de Nancy, Institut Élie Cartan, Nancy, 1984 | MR | Zbl

[2] H. Azad, A. Huckleberry, W. Richthofer, “Homogeneous CR-manifolds”, J. Reine Angew. Math., 358 (1985), 125–154 | MR | Zbl

[3] A. Huckleberry, W. Richthofer, “Recent developments in homogeneous CR-hypersurfaces”, Contributions to several complex variables, Aspects Math., E9, Vieweg, Braunschweig, 1986, 149–177 | MR | Zbl

[4] W. Richthofer, Homogene $\mathrm{CR}$-Mannigfaltigkeiten, Ph.D. Dissertation, Ruhr-Universität Bochum, 1985 | Zbl

[5] A. V. Loboda, “Homogeneous strictly pseudoconvex hypersurfaces in $\mathbb C^3$ with two-dimensional isotropy groups”, Sb. Math., 192:12 (2001), 1741–1761 | DOI | MR | Zbl

[6] A. V. Isaev, N. G. Kruzhilin, “Effective actions of $\mathrm{SU}_n$ on complex $n$-dimensional manifolds”, Illinois J. Math., 48:1 (2004), 37–57 | MR | Zbl

[7] A. Isaev, Lectures on the automorphism groups of Kobayashi-hyperbolic manifolds, 1902, Springer-Verlag, Lecture Notes in Math., 2007 | DOI | MR | Zbl

[8] G. Fels, W. Kaup, “CR-manifolds of dimension 5: a Lie algebra approach”, J. Reine Angew. Math., 604 (2007), 47–71 | DOI | MR | Zbl

[9] A. Morimoto, T. Nagano, “On pseudo-conformal transformations of hypersurfaces”, J. Math. Soc. Japan, 15 (1963), 289–300 | MR | Zbl

[10] T. Nagano, “Transformation groups with $(n-1)$-dimensional orbits on non-compact manifolds”, Nagoya Math. J., 14 (1959), 25–38 | MR | Zbl

[11] H. Rossi, “Homogeneous strongly pseudoconvex hypersurfaces” (Proc. Conf., Rice Univ., Houston, 1972), Rice Univ. Studies, 59:1 (1973), 131–145 | MR | Zbl

[12] D. V. Alekseevsky, A. F. Spiro, “Compact homogeneous CR manifolds”, J. Geom. Anal., 12:2 (2002), 183–201 | DOI | MR | Zbl

[13] A. Andreotti, G. A. Fredricks, “Embeddability of real analytic Cauchy–Riemann manifolds”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 6:2 (1979), 285–304 | MR | Zbl

[14] S. J. Greenfield, “Cauchy–Riemann equations in several variables”, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 275–314 | MR | Zbl

[15] Sh. Kobayashi, K. Nomizu, Foundations of differential geometry, vol. II, Intersci. Publ., New York–London–Sydney, 1969 | MR | MR | Zbl | Zbl

[16] C. Chevalley, “On the topological structure of solvable groups”, Ann. of Math. (2), 42:3 (1941), 668–675 | DOI | MR | Zbl

[17] M. Goto, “Faithful representations of Lie groups. I”, Math. Japonicae, 1 (1948), 107–119 | MR | Zbl

[18] W. Kaup, D. Zaitsev, “On local CR-transformation of Levi-degenerate group orbits in compact Hermitian symmetric spaces”, J. Eur. Math. Soc. (JEMS), 8:3 (2006), 465–490 | MR | Zbl

[19] D. Montgomery, “Simply connected homogeneous spaces”, Proc. Amer. Math. Soc., 1:4 (1950), 467–469 | DOI | MR | Zbl

[20] A. Borel, R. Remmert, “Über kompakte homogene Kählersche Mannigfaltigkeiten”, Math. Ann., 145:5 (1962), 429–439 | DOI | MR | Zbl

[21] É. B. Vinberg, B. N. Kimel'fel'd, “Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups”, Funct. Anal. Appl., 12:3 (1978), 168–174 | DOI | MR | Zbl

[22] A. Borel, Linear algebraic groups, Benjamin, New York–Amsterdam, 1969 | MR | MR | Zbl | Zbl

[23] A. T. Huckleberry, T. Wurzbacher, “Multiplicity-free complex manifolds”, Math. Ann., 286:1–3 (1990), 261–280 | DOI | MR | Zbl

[24] V. Guillemin, Sh. Sternberg, Symplectic techniques in physics, Cambridge Univ. Press, Cambridge, 1984 | MR | Zbl

[25] D. N. Akhiezer, “Actions with a finite number of orbits”, Funct. Anal. Appl., 19:1 (1985), 1–4 | DOI | MR | Zbl

[26] D. N. Akhiezer, Spherical varieties, Schriftenreihe, Heft Nr. 199, Ruhr-Universität–Bochum, 1993

[27] M. Brion, D. Luna, Th. Vust, “Espaces homogènes sphériques”, Invent. Math., 84:3 (1986), 617–632 | DOI | MR | Zbl

[28] M. Brion, “Spherical varieties: an introduction”, Topological methods in algebraic transformation groups (New Brunswick, NJ, 1988), Progr. Math., 80, Birkhäuser, Boston, MA, 1989, 11–26 | MR | Zbl

[29] D. Luna, Th. Vust, “Plongements d'espaces homogènes”, Comment. Math. Helv., 58:1 (1983), 186–245 | DOI | MR | Zbl

[30] J. E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, 21, Springer-Verlag, New York–Heidelberg–Berlin, 1975 | MR | MR | Zbl | Zbl

[31] M. Krämer, “Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen”, Compositio Math., 38:2 (1979), 129–153 | MR | Zbl

[32] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure Appl. Math., 80, Academic Press, New York–San Francisco–London, 1978 | MR | Zbl

[33] A. L. Onishchik, “Inclusion relations among transitive compact transformation groups”, Amer. Math. Soc. Transl. Ser. 2, 50 (1966), 5–58 | MR | Zbl | Zbl

[34] A. L. Onishchik, Topology of transitive transformation groups, Barth, Leipzig, 1994 | MR | MR | Zbl | Zbl

[35] G. Fels, A. Huckleberry, J. A. Wolf, Cycle spaces of flag domains. A complex geometric viewpoint, Progr. Math., 245, Birkhäuser, Boston, MA, 2006 | MR | Zbl

[36] T. Matsuki, “Orbits on affine symmetric spaces under the action of parabolic subgroups”, Hiroshima Math. J., 12:2 (1982), 307–320 | MR | Zbl

[37] T. Matsuki, “Double coset decompositions of reductive Lie groups arising from two involutions”, J. Algebra, 197:1 (1997), 49–91 | DOI | MR | Zbl

[38] K. Grove, B. Wilking, W. Ziller, “Positively curved cohomogeneity one manifolds and 3-Sasakian geometry”, J. Differential Geom., 78:1 (2008), 33–111 | MR | Zbl

[39] P. Heinzner, A. Huckleberry, Complex geometry of Hamiltonian actions, Springer-Verlag, 2009 (to appear)

[40] H. Azad, J. J. Loeb, “Plurisubharmonic functions and the Kempf–Ness theorem”, Bull. London Math. Soc., 25:2 (1993), 162–168 | DOI | MR | Zbl

[41] F. Berteloot, K. Oeljeklaus, “Invariant plurisubharmonic functions and hypersurfaces on semisimple complex Lie groups”, Math. Ann., 281:3 (1988), 513–530 | DOI | MR | Zbl