Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 081, 17 pages      arXiv:1003.1773     https://doi.org/10.3842/SIGMA.2010.081

Singular Reduction of Generalized Complex Manifolds

Timothy E. Goldberg
Donald and Helen Schort School of Mathematics and Computing Sciences, Lenoir-Rhyne University, Hickory, North Carolina 28601, USA

Received March 24, 2010, in final form October 06, 2010; Published online October 09, 2010

Abstract
In this paper, we develop results in the direction of an analogue of Sjamaar and Lerman's singular reduction of Hamiltonian symplectic manifolds in the context of reduction of Hamiltonian generalized complex manifolds (in the sense of Lin and Tolman). Specifically, we prove that if a compact Lie group acts on a generalized complex manifold in a Hamiltonian fashion, then the partition of the global quotient by orbit types induces a partition of the Lin-Tolman quotient into generalized complex manifolds. This result holds also for reduction of Hamiltonian generalized Kähler manifolds.

Key words: generalized complex manifold; Hamiltonian action; generalized complex quotient; Lin-Tolman quotient; singular reduction.

pdf (305 kb)   ps (212 kb)   tex (21 kb)

References

  1. Baird T., Lin Y., Topology of generalized complex quotients, J. Geom. Phys. 60 (2010), 1539-1557, arXiv:0802.1341.
  2. Bates L., Lerman E., Proper group actions and symplectic stratified spaces, Pacific J. Math. 181 (1997), 201-229, dg-ga/9407003.
  3. Ben-Bassat O., Boyarchenko M., Submanifolds of generalized complex manifolds, J. Symplectic Geom. 2 (2004), 309-355, math.DG/0309013.
  4. Bursztyn H., Cavalcanti G.R., Gualtieri M., Reduction of Courant algebroids and generalized complex structures, Adv. Math. 211 (2007), 726-765, math.DG/0509640.
  5. Courant T.J., Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), 631-661.
  6. Cushman R., Sniatycki J., Differential structure of orbit spaces, Canad. J. Math. 53 (2001), 715-755.
  7. Duistermaat J.J., Kolk J.A.C., Lie groups, Universitext, Springer-Verlag, Berlin, 2000.
  8. Goldberg T.E., Hamiltonian actions in integral Kähler and generalized complex geometry, PhD Thesis, Cornell University, 2010.
  9. Gualtieri M., Generalized complex structures, PhD Thesis, University of Oxford, 2003, math.DG/0401221.
  10. Guillemin V.W., Sternberg S., Supersymmetry and equivariant de Rham theory, Mathematics Past and Present, Springer-Verlag, Berlin, 1999.
  11. Hitchin N., Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), 281-308, math.DG/0209099.
  12. Hu S., Hamiltonian symmetries and reduction in generalized geometry, Houston J. Math. 35 (2009), 787-811, math.DG/0509060.
  13. Jotz M., Ratiu T.S., Sniatycki J., Singular reduction of Dirac structures, arXiv:0901.3062.
  14. Lin Y., Tolman S., Symmetries in generalized Kähler geometry, Comm. Math. Phys. 268 (2006), 199-222, math.DG/0509069.
  15. Marsden J., Weinstein A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121-130.
  16. Meyer K.R., Symmetries and integrals in mechanics, in Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, 259-272.
  17. Nitta Y., Convexity properties of generalized moment maps, J. Math. Soc. Japan 61 (2009), 1171-1204, arXiv:0901.0361.
  18. Ortega J.-P., Ratiu T.S., Momentum maps and Hamiltonian reduction, Progress in Mathematics, Vol. 222, Birkhäuser Boston, Inc., Boston, MA, 2004.
  19. Sjamaar R., Lerman E., Stratified symplectic spaces and reduction, Ann. of Math. (2) 134 (1991), 375-422.
  20. Stiénon M., Xu P., Reduction of generalized complex structures, J. Geom. Phys. 58 (2008), 105-121, math.DG/0509393.

Previous article   Next article   Contents of Volume 6 (2010)