Hamiltonian torus actions on symplectic orbifolds and toric varieties
HTML articles powered by AMS MathViewer
- by Eugene Lerman and Susan Tolman PDF
- Trans. Amer. Math. Soc. 349 (1997), 4201-4230 Request permission
Abstract:
In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly, we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each open facet and that all such orbifolds are algebraic toric varieties.References
- M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1–15. MR 642416, DOI 10.1112/blms/14.1.1
- W. L. Baily, On the imbedding of $V$-manifolds in projective space, Amer. J. Math. 79 (1957), 403–430. MR 100104, DOI 10.2307/2372689
- L. Bates and E. Lerman, Proper group actions and symplectic stratified spaces ( arXiv:dg-ga/9407003), to appear in Pacific J. Math.
- Kung-ching Chang, Infinite-dimensional Morse theory and multiple solution problems, Progress in Nonlinear Differential Equations and their Applications, vol. 6, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1196690, DOI 10.1007/978-1-4612-0385-8
- Thomas Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315–339 (French, with English summary). MR 984900, DOI 10.24033/bsmf.2100
- Hans Duistermaat, Victor Guillemin, Eckhard Meinrenken, and Siye Wu, Symplectic reduction and Riemann-Roch for circle actions, Math. Res. Lett. 2 (1995), no. 3, 259–266. MR 1338785, DOI 10.4310/MRL.1995.v2.n3.a3
- Mark Goresky and Robert MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. MR 932724, DOI 10.1007/978-3-642-71714-7
- V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491–513. MR 664117, DOI 10.1007/BF01398933
- V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515–538. MR 664118, DOI 10.1007/BF01398934
- André Haefliger and Éliane Salem, Actions of tori on orbifolds, Ann. Global Anal. Geom. 9 (1991), no. 1, 37–59. MR 1116630, DOI 10.1007/BF02411354
- P. Heinzner and F. Loose, Reduction of complex Hamiltonian $G$-spaces, Geom. Funct. Anal. 4 (1994), no. 3, 288–297. MR 1274117, DOI 10.1007/BF01896243
- Katsuo Kawakubo, The theory of transformation groups, Translated from the 1987 Japanese edition, The Clarendon Press, Oxford University Press, New York, 1991. MR 1150492
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- Serge Lang, Differential manifolds, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1972. MR 0431240
- E. Lerman, E. Meinrenken, S. Tolman and C. Woodward, Nonabelian convexity by symplectic cuts, arXiv:dg-ga/9603015.
- Eugene Lerman and Reyer Sjamaar, Reductive group actions on Kähler manifolds, Conservative systems and quantum chaos (Waterloo, ON, 1992) Fields Inst. Commun., vol. 8, Amer. Math. Soc., Providence, RI, 1996, pp. 85–92. MR 1383841
- Jean Mawhin and Michel Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989. MR 982267, DOI 10.1007/978-1-4757-2061-7
- E. Meinrenken, Symplectic surgery and Spin$^c$-Dirac operator, arXiv:dg-ga/9504002, to appear in Adv. in Math..
- Ichirô Satake, The Gauss-Bonnet theorem for $V$-manifolds, J. Math. Soc. Japan 9 (1957), 464–492. MR 95520, DOI 10.2969/jmsj/00940464
- Gerald W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63–68. MR 370643, DOI 10.1016/0040-9383(75)90036-1
- Reyer Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math. (2) 141 (1995), no. 1, 87–129. MR 1314032, DOI 10.2307/2118628
- Reyer Sjamaar and Eugene Lerman, Stratified symplectic spaces and reduction, Ann. of Math. (2) 134 (1991), no. 2, 375–422. MR 1127479, DOI 10.2307/2944350
Additional Information
- Eugene Lerman
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- Address at time of publication: Department of Mathematics, Unversity of Illinois at Urbana-Champaign, 1409 West Green St., Urbana, Illinois 61801
- Email: lerman@math.uiuc.edu
- Susan Tolman
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; Current address $\mathrm {(both authors)}$: $\mathrm {Department of Mathematics, University of Illinois at Urbana}$ -$\mathrm {Champaign, 1409 West Green St., Urbana, Illinois 61801}$
- Address at time of publication: Department of Mathematics, Unversity of Illinois at Urbana-Champaign, 1409 West Green St., Urbana, Illinois 61801
- Email: tolman@math.princeton.edu
- Received by editor(s): July 27, 1995
- Received by editor(s) in revised form: April 22, 1996
- Additional Notes: Both authors were partially supported by NSF postdoctoral fellowships.
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 349 (1997), 4201-4230
- MSC (1991): Primary 58F05; Secondary 57S15, 14M25
- DOI: https://doi.org/10.1090/S0002-9947-97-01821-7
- MathSciNet review: 1401525