Complex-symmetric spaces

Ralf Lehmann

Annales de l'institut Fourier (1989)

  • Volume: 39, Issue: 2, page 373-416
  • ISSN: 0373-0956

Abstract

top
A compact complex space X is called complex-symmetric with respect to a subgroup G of the group Aut 0 ( X ) , if each point of X is isolated fixed point of an involutive automorphism of G . It follows that G is almost G 0 -homogeneous. After some examples we classify normal complex-symmetric varieties with G 0 reductive. It turns out that X is a product of a Hermitian symmetric space and a compact torus embedding satisfying some additional conditions. In the smooth case these torus embeddings are classified using the description of torus embeddings by systems of cone (“fans”) and the theory of Coxeter groups.

How to cite

top

Lehmann, Ralf. "Complex-symmetric spaces." Annales de l'institut Fourier 39.2 (1989): 373-416. <http://eudml.org/doc/74835>.

@article{Lehmann1989,
abstract = {A compact complex space $X$ is called complex-symmetric with respect to a subgroup $G$ of the group $\{\rm Aut\}_0(X)$, if each point of $X$ is isolated fixed point of an involutive automorphism of $G$. It follows that $G$ is almost $G^0$-homogeneous. After some examples we classify normal complex-symmetric varieties with $G^0$ reductive. It turns out that $X$ is a product of a Hermitian symmetric space and a compact torus embedding satisfying some additional conditions. In the smooth case these torus embeddings are classified using the description of torus embeddings by systems of cone (“fans”) and the theory of Coxeter groups.},
author = {Lehmann, Ralf},
journal = {Annales de l'institut Fourier},
keywords = {almost-homogeneous spaces; almost-homogeneous varieties; fans; holomorphic involutions; spherical varieties; toric varieties; torus embeddings; Coxeter groups},
language = {eng},
number = {2},
pages = {373-416},
publisher = {Association des Annales de l'Institut Fourier},
title = {Complex-symmetric spaces},
url = {http://eudml.org/doc/74835},
volume = {39},
year = {1989},
}

TY - JOUR
AU - Lehmann, Ralf
TI - Complex-symmetric spaces
JO - Annales de l'institut Fourier
PY - 1989
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 2
SP - 373
EP - 416
AB - A compact complex space $X$ is called complex-symmetric with respect to a subgroup $G$ of the group ${\rm Aut}_0(X)$, if each point of $X$ is isolated fixed point of an involutive automorphism of $G$. It follows that $G$ is almost $G^0$-homogeneous. After some examples we classify normal complex-symmetric varieties with $G^0$ reductive. It turns out that $X$ is a product of a Hermitian symmetric space and a compact torus embedding satisfying some additional conditions. In the smooth case these torus embeddings are classified using the description of torus embeddings by systems of cone (“fans”) and the theory of Coxeter groups.
LA - eng
KW - almost-homogeneous spaces; almost-homogeneous varieties; fans; holomorphic involutions; spherical varieties; toric varieties; torus embeddings; Coxeter groups
UR - http://eudml.org/doc/74835
ER -

References

top
  1. [A] D. N. AHIEZER, On Algebraic Varieties that are Symmetric in the Sense of Borel, Sov. Math. Dokl., 30 (1984), 579-582. Zbl0589.32052
  2. [BB] A. BIALYNICKI-BIRULA, Some Theorems on Actions of Algebraic Groups, Ann. Math., 98 (1973), 480-497. Zbl0275.14007MR51 #3186
  3. [Bo] A. BOREL, Symmetric Compact Complex Spaces. Arch. Math., 33 (1979), 49-56. Zbl0423.32015MR80k:32033
  4. [B-L-V] M. BRION, D. LUNA, T. VUST, Espaces homogènes sphériques, Inv. Math., 84 (1986), 617-632. Zbl0604.14047MR87g:14057
  5. [B-P] M. BRION, F. PAUER, Valuations des espaces homogènes sphériques, Comm. Math. Helv., 62 (1987), 265-285. Zbl0627.14038MR88h:14051
  6. [Ca] E. CARTAN, Œuvres Complètes, Gauthier-Villars, Paris, 1952. 
  7. [Car] H. CARTAN, Quotients of Complex Analytic Spaces, Contributions to Function Theory, Bombay, 1960. Zbl0122.08702MR25 #3199
  8. [Cox] H. S. M. COXETER, Discrete Groups Generated by Reflections, Ann. Math., 35 (1934), 588-621. Zbl0010.01101JFM60.0898.02
  9. [Dan] V. I. DANILOV, The Geometry of Toric Varieties, Russian Math. Surveys, 33 (1978), 97-154. Zbl0425.14013MR80g:14001
  10. [EGA] A. GROTHENDIECK, J. A. DIEUDONNÉ, Éléments de Géométrie Algébrique, Presses Universitaires de France, Paris, 1967. 
  11. [G-B] L. C. GROVE, C. T. BENSON, Finite Reflection Groups. Graduate Texts in Mathematics, 99, Springer, New York, 1985. Zbl0579.20045MR85m:20001
  12. [G-R] R. C. GUNNING, H. ROSSI, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. Zbl0141.08601MR31 #4927
  13. [Hel] S. HELGASON, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962. Zbl0111.18101MR26 #2986
  14. [Ho] H. HOLMANN, Komplexe Räume mit komplexen Transformationsgruppen, Math. Ann., 150 (1963), 327-359. Zbl0156.30603MR27 #776
  15. [H-O] A. T. HUCKLEBERRY, E. OELJEKLAUS, Classification Theorems for Almost Homogeneous Spaces, Institut Elie Cartan, 9 (1984). Zbl0549.32024MR86g:32050
  16. [Jän] K. JÄNICH, Differenzierbare G-Mannigfaltigkeiten, Springer, Berlin, 1968. Zbl0159.53701
  17. [K] W. KAUP, Bounded Symmetric Domains in Finite and Infinite Dimensions, Several Complex Variables, Cortona (1976-1977), 180-191. Zbl0418.32020MR84b:32045
  18. [Ko] J. KONARSKI, Decompositions of Normal Algebraic Varieties Determined by an Action of a One-dimensional Torus, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 26 (1978), 295-300. Zbl0394.14019MR81b:14026
  19. [Kr] H. KRAFT, Geometrische Methoden in der Invariantentheorie, Vieweg, Braunschweig, 1984. Zbl0569.14003MR86j:14006
  20. [L1] R. LEHMANN, Complex-symmetric Spaces. Dissertation, Bochum, 1988. Zbl0689.14025
  21. [L2] R. LEHMANN, Complex-symmetric Torus Embeddings. Schriftenreihe d. Fachbereichs Mathematik der Universität Duisburg, 126 (1987). 
  22. [L-V] D. LUNA, T. VUST, Plongements d'espaces homogènes, Comm. Math. Helv., 58 (1983), 186-245. Zbl0545.14010MR85a:14035
  23. [Ma] Y. MATSUSHIMA, Fibrés holomorphes sur un tore complexe, Nag. Math. J., 14 (1959), 1-24. Zbl0095.36702MR21 #1403
  24. [Mon] D. MONTGOMERY, Simply Connected Homogeneous Spaces, Proc. Am. Math. Soc., 1 (1950), 467-469. Zbl0041.36309MR12,242c
  25. [Mo] G. D. MOSTOW, Self-adjoint Groups, Ann. Math., (2), 62 (1955), 44-55. Zbl0065.01404MR16,1088a
  26. [Oda] T. ODA, Lectures on Torus Embeddings and Applications. Tata Institute of Fundamental Research, Bombay, 1978. Zbl0417.14043MR81e:14001
  27. [P] J. POTTERS, On Almost Homogeneous Compact Complex Surfaces, Inv. Math., 8 (1969), 244-266. Zbl0205.25102MR41 #3808
  28. [Su] H. SUMIHIRO, Equivariant Completion. J. Math. Kyoto Univ., 14-1 (1974), 1-28. Zbl0277.14008MR49 #2732
  29. [T-E] G. KEMPF, F. KNUDSEN, D. MUMFORD, B. SAINT-DONAT, Toroidal Embeddings I, Lecture Notes in Mathematics, 339, Springer, Berlin, 1973. Zbl0271.14017MR49 #299
  30. [V] T. VUST, Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France, 102 (1974), 317-333. Zbl0332.22018MR51 #3187

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.