symplectic orbifolds and toric varieties ]]> Hamiltonian torus actions on symplectic orbifolds and toric varieties Eugene Lerman Eugene Lerman lerman@math.uiuc.edu Susan Tolman tolman@math.princeton.edu Lerman_Eugene | Tolman_Susan ]]> 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. A A M. F. Atiyah, Convexity and commuting hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15. 83e:53037 Ba Baily W. L. Baily, On the imbedding of V -manifolds in projective space, Amer. J. Math. 79 (1957), 403--430. 20:6538 BL BL L. Bates and E. Lerman, Proper group actions and symplectic stratified spaces ( dg-ga 9407003 ), to appear in Pacific J. Math. C Chang K.-C. Chang, Infinite dimensional Morse theory and multiple solution problems , Boston, Birkhauser, 1993. 94e:58023 D Del T. Delzant, Hamiltoniens periodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), 315-339. 90b:58069 DGMW DGMW J.J. Duistermaat, V. Guillemin, E. Meinrenken and S. Wu, Symplectic reduction and Riemann-Roch for circle actions, Math. Research Letters 2 (1995), 259--266. 96g:58178 GM MTSS M. Goresky and R. MacPherson, Stratified Morse Theory Berlin, New York: Springer-Verlag, 1988. 90d:57039 GS1 GS V. Guillemin and S. Sternberg, Convexity properties of the moment mapping I, Invent. Math. 67 (1982), 491-513. 83m:58037 GS2 GS:gq V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515--538. 83m:58040 HS HS A. Haefliger and E. Salem, Actions of tori on orbifolds, Ann. Global Anal. Geom. 9 (1991), 37--59. 92f:57047 HL HL P, Heinzner and F. Loose, Reduction of Hamiltonian G -spaces, Geom. and Func. Analysis 4 (1994), 288--297. 95j:58050 K Kaw K. Kawakubo, The Theory of Transformation Groups , Oxford University Press, 1991. 93g:57044 Ko Kosz J.L. Koszul, Sur certains groupes de transformations de Lie, dans Colloque de Geometrie Differentielle, Colloques du CNRS 71 (1953), 137--141. 15:600g La Lang S. Lang, Differential manifolds , Reading, Mass., Addison-Wesley Pub. Co., 1972. 55:4241 LMTW LMTW E. Lerman, E. Meinrenken, S. Tolman and C. Woodward, Nonabelian convexity by symplectic cuts, dg-ga 9603015. LS LS E. Lerman and S. Sjamaar, Reductive Group Actions on Kaehler Manifolds, in Conservative Systems and Quantum Chaos , Fields Inst. Comm., vol. 8, Amer. Math. Soc., Providence, 1996, pp. 85--92. 97c:32045 MW MW J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems , New York : Springer-Verlag, 1989. 90e:58016 M Meinrenken E. Meinrenken, Symplectic surgery and Spin c -Dirac operator, dg-ga 9504002 , to appear in Adv. in Math. . S Sa I. Satake, The Gauss-Bonnet theorem for V -manifolds J. Math. Soc. Japan 9 (1957), 464--492. 20:2022 Sch sc:sm G. W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63--68. 51:6870 Sj Sj R. Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. Math. 141 (1995), 87--129. 96a:58098 SL SL R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction Ann. Math. 134 (1991), 375--422. 92g:58036 1997 American Mathematical Society Transactions of the American Mathematical Society Trans. Amer. Math. Soc. tran 349 10 19971001 July 27, 1995 April 22, 1996 4201 4201-4230 30 S0002-9947-97-01821-7 10.1090/S0002-9947-97-01821-7 0002-9947 1088-6850 English 58F05 57S15 14M25 1991 http://www.ams.org/jourcgi/jour-getitem?pii=S0002-9947-97-01821-7