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Polar and coisotropic actions on Kähler manifolds

Author(s): Fabio Podestà; Gudlaugur Thorbergsson
Journal: Trans. Amer. Math. Soc. 354 (2002), 1759-1781.
MSC (2000): Primary 53C55, 57S15
Posted: January 10, 2002
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Abstract: The main result of the paper is that a polar action on a compact irreducible homogeneous Kähler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.

References:

[Ab]
M. Abreu, Kähler geometry of toric manifolds in symplectic coordinates, Preprint, DG/0004122.

[AA]
A. Alekseevsky and D. Alekseevsky, Asystatic $G$-manifolds, Proceedings of the Workshop on Differential Geometry and Topology, Eds. R. Caddeo, F. Tricerri, World Scientific, Singapore, 1993, pp. 1-22. MR 96f:57035

[BR]
C. Benson and G. Ratcliff, A classification of multiplicity free actions, J. Algebra 181 (1996), 152-186. MR 97c:14046

[Bl]
A. Blanchard, Les variété analytique complexes, Ann. Ec. Norm. 73 (1956), 157-202. MR 19:316e

[BH]
R. Blumenthal and J. Hebda, De Rham decomposition theorems for foliated manifolds, Ann. Inst. Fourier Grenoble 33 (1983), 183-198. MR 84j:53042

[Bo]
A. Borel, Le plan projectif des octaves et les sphères comme espaces homogènes, C. R. Acad. Sci. Paris 230 (1950), 1378-1380. MR 11:640c

[BS]
R. Bott and H. Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964-1029. Correction in Amer. J. Math. 83 (1961), 207-208. Also in the Collected Papers of Raoul Bott, Volume 1, 327-393 and 503-505, Birkhäuser, Boston, 1994. MR 21:4430; MR 30:589; MR 95i:01027

[Br1]
M. Brion, Quelques propriétés des espaces homogènes sphériques, Manuscripta Math. 55 (1986), 191-198. MR 87g:14054

[Br2]
M. Brion, Classification des espaces homogènes sphérique, Compositio Math. 63 (1987), 189-208. MR 89d:32065

[BtD]
T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, Berlin-New York, 1985. MR 86i:22023

[Co]
L. Conlon, Variational completeness and K-transversal domains, J. Differential Geom. 5 (1971), 135-147. MR 45:4320

[Da]
J. Dadok, Polar coordinates induced by actions of compact Lie groups, Trans. A.M.S. 288 (1985), 125-137. MR 86k:22019

[De]
Th. Delzant, Hamiltoniens périodiques et images convexes de l'applicationn moment, Bull. Soc. math. France 116 (1988), 315-339. MR 90b:58069

[Du]
J.J. Duistermaat, Convexity and tigthness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution, Trans. A.M.S. 275 (1983), 417-429. MR 84c:53035

[EH]
J.-H. Eschenburg and E. Heintze, On the classification of polar representations, Math. Z. 232 (1999), 391-398. MR 2001g:53099

[Fr]
T. Frankel, Fixed points on Kähler manifolds, Ann. of Math. 70 (1959), 1-8. MR 24:A1730

[GT]
C. Gorodski and G. Thorbergsson, Representations of compact Lie groups and the osculating spaces of their orbits, Preprint.

[GS1]
V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, II, Invent. Math. 77 (1984), 533-546. MR 86b:58042a

[GS2]
V. Guillemin and S. Sternberg, Multiplicity-free spaces, J. Differential Geometry 19 (1984), 31-56. MR 85h:58071

[GS3]
V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambridge University Press, 1984. MR 86f:58054

[HPTT]
E. Heintze, R. Palais, C. L. Terng and G. Thorbergsson, Hyperpolar actions and k-flat homogeneous spaces, J. reine angew. Math. 454 (1994), 163-179. MR 96b:53062

[HW]
A.T. Huckleberry and T. Wurzbacher, Multiplicity-free complex manifolds, Math. Annalen 286 (1990), 261-280. MR 91c:32027

[Ka]
V.G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), 190-213. MR 81i:17005

[Ki]
B. Kimelfeld, Homogeneous domains on flag manifolds, J. Math. Anal. Appl. 121 (1987), 506-588. MR 88h:32027

[Kir]
F. C. Kirwan, Convexity properties of the momentum mapping, III, Invent. Math. 77 (1984), 547-552. MR 86b:58042b

[KN]
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. I, Interscience Publishers, J. Wiley & Sons, 1963. MR 27:2945

[Ko]
A. Kollross, A classification of hyperpolar and cohomogeneity one actions, PhD thesis, Augsburg (1998).

[Kr]
H. Kraft, Geometrische Methoden in der Invariantentheorie, Vieweg, Braunschweig, Wiesbaden, 1984. MR 86j:14006

[Kra]
M. Krämer, Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compos. Math. 38 (1979), 129-153. MR 80f:22011

[Le]
A. Leahy, A classification of multiplicity free representations, J. Lie Theory 8 (1998), 367-391. MR 2000g:22024

[LE]
S. Lie and F. Engel, Theorie der Transformationsgruppen, vol. I, Teubner, Leipzig, 1888.

[On]
A. L. Onishchik, Topology of transitive transformation groups, Johann Ambrosius, Barth, Leipzig, Berlin, New York, 1994. MR 95e:57058

[PT1]
R.S. Palais and C.-L. Terng, A general theory of canonical forms, Trans. A.M.S 300 (1987), 771-789. MR 88f:57069

[PT2]
R.S. Palais and C.-L. Terng, Critical point theory and submanifold geometry, Lecture Notes in Mathematics 1353, Springer-Verlag, Berlin-New York, 1988. MR 90c:53143

[PTh]
F. Podestà and G. Thorbergsson, Polar actions on rank one symmetric spaces, J. Differential Geom. 53 (1999), 131-175. CMP 2000:16

[Sh]
A.N. Shchetinin, On a class of compact homogeneous spaces I, Ann. Global Anal. Geom. 6 (1988), 119-140. MR 90d:57049

[Ya]
O. Yasukura, A classification of orthogonal transformation groups of low cohomogeneity, Tsukuba J. Math. 10 (1986), 299-326. MR 88b:57036

[Wo]
J.A. Wolf, Spaces of constant curvature, (Fourth edition), Publish or Perish, Berkeley, 1977. (Fifth ed., 1984. MR 88k:53002)

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Additional Information:

Fabio Podestà
Affiliation: Dipartimento di Matematica e Applicazioni per l'Architettura, Università di Firenze, Piazza Ghiberti 27, I-50142 Firenze, Italy
Email: podesta@math.unifi.it

Gudlaugur Thorbergsson
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany
Email: gthorbergsson@mi.uni-koeln.de

DOI: 10.1090/S0002-9947-02-02902-1
PII: S 0002-9947(02)02902-1
Keywords: Polar and coisotropic actions, homogeneous K\"{a}hler manifolds
Received by editor(s): November 8, 2000
Received by editor(s) in revised form: July 31, 2001
Posted: January 10, 2002
Additional Notes: Part of the work on this paper was done during a visit of the second author to the University of Florence and was financially supported by G.N.S.A.G.A. - I.N.d.A.M
Copyright of article: Copyright 2002, American Mathematical Society

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