On invariant domains in certain complex homogeneous spaces

Xiang-Yu Zhou

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 4, page 1101-1115
  • ISSN: 0373-0956

Abstract

top
Given a compact connected Lie group K . For a relatively compact K -invariant domain D in a Stein K ℂ -homogeneous space, we prove that the automorphism group of D is compact and if K is semisimple, a proper holomorphic self mapping of D is biholomorphic.

How to cite

top

Zhou, Xiang-Yu. "On invariant domains in certain complex homogeneous spaces." Annales de l'institut Fourier 47.4 (1997): 1101-1115. <http://eudml.org/doc/75256>.

@article{Zhou1997,
abstract = {Given a compact connected Lie group $K$. For a relatively compact $K$-invariant domain $D$ in a Stein $K^\{\Bbb C\}$-homogeneous space, we prove that the automorphism group of $D$ is compact and if $K$ is semisimple, a proper holomorphic self mapping of $D$ is biholomorphic.},
author = {Zhou, Xiang-Yu},
journal = {Annales de l'institut Fourier},
keywords = {Stein homogeneous spaces; automorphism groups; proper holomorphic mappings; invariant domains},
language = {eng},
number = {4},
pages = {1101-1115},
publisher = {Association des Annales de l'Institut Fourier},
title = {On invariant domains in certain complex homogeneous spaces},
url = {http://eudml.org/doc/75256},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Zhou, Xiang-Yu
TI - On invariant domains in certain complex homogeneous spaces
JO - Annales de l'institut Fourier
PY - 1997
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 4
SP - 1101
EP - 1115
AB - Given a compact connected Lie group $K$. For a relatively compact $K$-invariant domain $D$ in a Stein $K^{\Bbb C}$-homogeneous space, we prove that the automorphism group of $D$ is compact and if $K$ is semisimple, a proper holomorphic self mapping of $D$ is biholomorphic.
LA - eng
KW - Stein homogeneous spaces; automorphism groups; proper holomorphic mappings; invariant domains
UR - http://eudml.org/doc/75256
ER -

References

top
  1. [1] A. ANDREOTTI, T. FRANKEL, The Lefschetz theorem on hyperplane sections, Ann. of Math., 69 (1959), 713-717. Zbl0115.38405MR31 #1685
  2. [2] H. AZAD, J.-J. LOEB, Plurisubharmonic functions and the Kempf-Ness theorem, Bull. London Math. Soc., 25 (1993), 162-168. Zbl0795.32002MR93m:32044
  3. [3] E. BEDFORD, Proper holomorphic mappings, Bull. Amer. Math. Soc., 10 (1984), 157-175. Zbl0534.32009MR85b:32041
  4. [4] E. BEDFORD, On the automorphism group of a Stein manifold, Math. Ann., 266 (1983), 215-227. Zbl0532.32014MR85h:32049
  5. [5] A. BOREL, Symmetric compact complex spaces, Arch. Math., 33 (1979), 49-56. Zbl0423.32015MR80k:32033
  6. [6] R. BOTT, L. TU, Differential forms in algebraic topology, Springer-Verlag, New York, Heidelberg, Berlin, 1982. Zbl0496.55001MR83i:57016
  7. [7] G. FELS, L. GEATTI, Invariant domains in complex symmetric spaces, J. reine angew. Math., 454 (1994), 97-118. Zbl0803.32019MR95g:32052
  8. [8] F. FORSTNERIČ, Proper holomorphic mappings. A survey, in Proceedings of the special year on several complex variables at Mittag-Leffler Institute, E. Fornaess ed., Princeton University Press, Princeton, 1993. Zbl0778.32008MR94a:32042
  9. [9] H. GRAUERT, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann., 135 (1958), 263-273. Zbl0081.07401MR20 #4661
  10. [10] W. GREUB, S. HALPERIN, R. VANSTONE, Connections, curvature, and cohomology, Academic Press, New York, London, 1972. 
  11. [11] P. HEINZNER, Geometric invariant theory on Stein space, Math. Ann., 289 (1991), 631-662. Zbl0728.32010MR92j:32116
  12. [12] P. HEINZNER, Equivariant holomorphic extensions of real analytic manifolds, Bull. Soc. Math. France., 121 (1993), 445-463. Zbl0794.32022MR94i:32050
  13. [13] P. HEINZNER, A. HUCKLEBERRY, Invariant plurisubharmonic exhaustions and retractions, Manu. Math., 83 (1994), 19-29. Zbl0842.32010MR95a:32055
  14. [14] P. HEINZNER, F. KUTZSCHEBAUCH, An equivariant version of Grauert's Oka principle, Invent. Math., 119 (1995), 317-346. Zbl0837.32004MR96c:32034
  15. [15] M. HIRSCH, Differential topology, Springer-Verlag, New York, Heidelberg, Berlin, 1976. Zbl0356.57001MR56 #6669
  16. [16] F. HIRZEBRUCH, Topological methods in algebraic geometry, Springer-Verlag, New York, Heidelberg, Berlin, 1966. Zbl0138.42001MR34 #2573
  17. [17] L. HÖRMANDER, An introduction to complex analysis in several variables, North-Holland, Amsterdam, 1966. Zbl0138.06203
  18. [18] D. HUSEMOLLER, Fibre bundles, McGraw-Hill, New York et al., 1966. Zbl0144.44804MR37 #4821
  19. [19] L. KAUP, B. KAUP, Holomorphic functions of several variables, Walter de Gruyter, Berlin, New York, 1983. Zbl0528.32001
  20. [20] S. KOBAYASHI, Intrinsic distances, measures, and geometric function theory, Bull. Amer. Math. Soc., 82 (1976), 357-416. Zbl0346.32031
  21. [21] H. KRAFT, Geomtrische Methoden in der Invariantentheri, Braunschweig-Wiesbaden, Vieweg, 1985. 
  22. [22] M. LASSALLE, Séries de Laurent des functions holomorphes dans la complexification d'un espace symétrique compact, Ann. Scient. Éc. Norm. Sup., 11 (1978), 167-210. Zbl0452.43011MR80f:22006
  23. [23] W. MASSEY, A basic course in algebraic topology, Springer-Verlag, New York, Heidelberg, Berlin, 1991. Zbl0725.55001MR92c:55001
  24. [24] J. MILNOR, On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc., 90 (1959), 272-280. Zbl0084.39002MR20 #6700
  25. [25] N. MOK, Rigiditity of holomorphic self-mappings and the automorphism groups of hyperbolic Stein spaces, Math. Ann., 266 (1984), 433-447. Zbl0574.32021MR85f:32044
  26. [26] R. NARASIMHAN, On the homology group of Stein spaces, Invent. Math., 2 (1967), 377-385. Zbl0148.32202MR35 #7356
  27. [27] R. NARASIMHAN, Several complex variables, University of Chicago, Chicago, 1971. Zbl0223.32001MR49 #7470
  28. [28] A.L. ONISHCHIK, Topology of transitive transformation groups, Fizmatlit Publishing Company, Moscow, 1994 (Russian). Zbl0796.57001MR95e:57058
  29. [29] S. PINCHUK, On proper holomorphic mappings of strictly psedoconvex domains, Siberian Math. J., 15 (1974), 909-917 (Russian). Zbl0303.32016
  30. [30] R. REMMERT, K. STEIN, Eigentlische holomorphe Abbildungen, Math. Z., 73 (1960), 159-189. Zbl0102.06903MR23 #A1840
  31. [31] N. STEENROD, Topology of fibre bundles, Princeton University Press, Princeton, 1951. Zbl0054.07103MR12,522b
  32. [32] V. VARADARAJAN, Lie groups, Lie algebras, and their representations, Springer-Verlag, New York, Heidelberg, Berlin, 1984. Zbl0955.22500
  33. [33] J. WOLF, Spaces of constant curvature, Publish or Perish, Boston, 1974. Zbl0281.53034MR49 #7958
  34. [34] X.Y. ZHOU, On orbit connectedness, orbit convexity, and envelopes of holomorphy, Izvestija RAN., Ser. Math., 58 (1994), 196-205. Zbl0835.32006

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.