Homogeneous complex manifolds satisfying various types of Levi conditions are considered. Classical results which were of particular interest to Andreotti are recalled. Convexity and concavity properties of flag domains are discussed in some detail. A precise classification of pseudoconvex flag domains is given. It is shown that flag domains which are in a certain sense generic are pseudoconcave.
@article{BUMI_2010_9_3_1_1_0, author = {Alan Huckleberry}, title = {Remarks on Homogeneous Complex Manifolds Satisfying Levi Conditions}, journal = {Bollettino dell'Unione Matematica Italiana}, volume = {3}, year = {2010}, pages = {1-23}, zbl = {1215.32015}, mrnumber = {2605911}, language = {en}, url = {http://dml.mathdoc.fr/item/BUMI_2010_9_3_1_1_0} }
Huckleberry, Alan. Remarks on Homogeneous Complex Manifolds Satisfying Levi Conditions. Bollettino dell'Unione Matematica Italiana, Tome 3 (2010) pp. 1-23. http://gdmltest.u-ga.fr/item/BUMI_2010_9_3_1_1_0/
[A] Théorèmes de dépendance algébrique sur les espaces complexes pseudo-concaves, Bull. Soc. Math. France, 91 (1963), 1-38. | MR 152674 | Zbl 0113.06403
,[AGh] Some remarks on quasi-abelian manifolds, Global analysis and its applications (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phy. Trieste, 1972), Vol. II. Internat. Atomc Energy Agency, Vienna (1974), 203-206. | MR 435460
- ,[AGr1] Algebraische Körper von automorphen Funktionen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl.II, 1961 (1961), 39-48. | MR 132211
- ,[AGr2] Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, 90 (1962), 193-259. | MR 150342 | Zbl 0106.05501
- ,[AH] Pseudoconcave Lie groups, Compositio Math., 25 (1972), 109-115. | MR 316752 | Zbl 0239.32018
- ,[AN1] La convexité holomorphe dans l'espace analytique des cycles d'une variété algébrique. Ann. Scuola Norm. Sup. Pisa (3), 21 (1967), 31-82. | MR 239118 | Zbl 0176.04001
- ,[AN2] Cycles of algebraic manifolds and -cohomology, Ann. Scuola Norm. Sup. Pisa (3), 25 (1971), 59-114. | MR 288314 | Zbl 0212.53701
- ,[AK] 1759, Springer-Verlag Berlin (2001). | MR 1836462 | Zbl 0974.22004
- , Line bundles, cohomology and quasi-abelian, Lecture Notes in Mathematics[Ba] Espace analytique réduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie, "Fonctions de plusieurs variables complexes", II (Sém. François Norguet, 1974-1975), Springer Lecture Notes in Math., 482 (1975), 1-158. | MR 399503
,[Be] Fontions plurisousharmoniques sur SL(2,C) invariantes par un sous-groupe monogène, J. Analyse Math., 48 (1987), 267-276. | MR 910012 | Zbl 0635.32018
,[BeO] Invariant plurisubharmonic functions and hypersurfaces on semisimple complex Lie groups, Math. Ann., 281, no. 3 (1988), 513-530. | MR 954156 | Zbl 0653.32029
- ,[Bo] Pseudo-concavité et groupes arithmétiques, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York (1970), 70-84. | MR 262547
,[C] Sur les fonctions périodique, Ann. Sci. École Norm. Sup. (3), 19 (1902), 9-61. | MR 1509004 | Zbl 33.0438.01
,[FHW] 245 (Springer/ Birkhäuser Boston, 2005). | MR 2188135 | Zbl 1084.22011
- - , Cycles Spaces of Flag Domains: A Complex Geometric Viewpoint, Progress in Mathematics, Volume[GH] On non-compact complex nil-manifolds, Math. Ann., 238 (1978), 39-49. | MR 510305 | Zbl 0405.32009
- ,[HI] Integration of local actions on holomlorphic fiber spaces, Nagoya Math. J., 146 (1997), 31-53. | MR 1460953 | Zbl 0879.55006
- ,[HoH] On closures of cycle spaces of flag domains, Manuscripta Math., 121 (2006), 317-327. | MR 2271422 | Zbl 1115.22011
- ,[HN] On k-pseudoflat complex spaces, Math. Ann., 200, 1 (1973), 1-10. | MR 324065 | Zbl 0237.32007
- ,[HS1] Pseudoconcave Homogeneous Manifolds, Ann. Scuola Norm. Sup. Pisa 7 (1980), 29-54. | MR 577324 | Zbl 0506.32015
- ,[HS2] A Classification of Strictly Pseudoconcave Homogeneous Manifolds, Ann. Scuola Norm. Sup. Pisa 8 (1981), 231-255. | MR 623936 | Zbl 0464.32019
- ,[HSt] On the Thickness of the Shilov boundary, Math. Ann., 207 (1974), 213-231. | MR 338445 | Zbl 0276.46026
- ,[HW] Cycle space constructions for exhaustions of flag domains (To appear in Ann. Scuola Norm. Pisa 93 (2010), arXiv:0807.2062). | MR 2722656 | Zbl 1209.32019
- ,[K] 29 (1964). | MR 166298
, Maximale Untergruppen Abelscher komplexer Liescher Gruppen, Schr. Math. Inst. Univ. Münster,[L] Action d'une forme réelle d'un groupe de Lie complex sur les fonctions plurisousharmoniques, Ann. Inst. Fourier (Grenoble), 35 , no. 4 (1985), 59-97. | MR 812319 | Zbl 0563.32013
,[LOR] A decomposition theorem for complex nilmanifolds, Canad. Math. Bull, 30, no. 3 (1987), 377-378. | MR 906364 | Zbl 0627.32023
- - ,[M] Espace homogèenes de Stein des groupes de Lie complexes, Nagoya Math. J., 16 (1960), 205-218. | MR 109854 | Zbl 0094.28201
,[MM] Sur certains espaces fibrés holomorphes sur une variété de Stein, Bull. Soc. Math. France, 88 (1960), 137-155. | MR 123739 | Zbl 0094.28104
- ,[Mo] Non-compact complex Lie groups without non-constant holomorphic functions, Proc. Conf. Complex Analysis (Minneapolis), Springer, Berlin (1964), 256-272. | MR 181702
,[S] Homogeneous complex manifolds and representations of semisimple Lie groups, thesis, University of California at Berkeley, 1967. | MR 2617034
,[SW] A vanishing theorem for open orbits on complex flag manifolds, Proc. Amer. Math. Soc., 92 (1984), 461-464. | MR 759674 | Zbl 0558.22010
- ,[W] The action of a real semisimple Lie group on a complex manifold, I: Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc., 75 (1969), 1121-1237. | MR 251246 | Zbl 0183.50901
,