Convexity properties of generalized moment maps
Yasufumi NITTA
Source: J. Math. Soc. Japan Volume 61, Number 4
(2009), 1171-1204.
Abstract
In this paper, we consider generalized moment maps for Hamiltonian actions on $H$-twisted generalized complex manifolds introduced by Lin and Tolman [15]. The main purpose of this paper is to show convexity and connectedness properties for generalized moment maps. We study Hamiltonian torus actions on compact $H$-twisted generalized complex manifolds and prove that all components of the generalized moment map are Bott-Morse functions. Based on this, we shall show that the generalized moment maps have a convex image and connected fibers. Furthermore, by applying the arguments of Lerman, Meinrenken, Tolman, and Woodward [13] we extend our results to the case of Hamiltonian actions of general compact Lie groups on $H$-twisted generalized complex orbifolds.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1257520504
Digital Object Identifier: doi:10.2969/jmsj/06141171
Zentralblatt MATH identifier: 05651148
Mathematical Reviews number (MathSciNet): MR2588508
References
M. F. Atiyah, Convexity and commuting hamiltonians, Bull. London Math. Soc., 14 (1982), 1–15.
Mathematical Reviews (MathSciNet): MR642416
Zentralblatt MATH: 0482.58013
Digital Object Identifier: doi:10.1112/blms/14.1.1
O. Ben-Bassat and M. Boyarchenko, Submanifolds of generalized complex manifolds, J. Symp. Geom., 2 (2004), 309–355.
Mathematical Reviews (MathSciNet): MR2131639
Zentralblatt MATH: 1082.53077
Project Euclid: euclid.jsg/1118755324
W. M. Boothby, S. Kobayashi and H. C. Wang, A note on mappings and automorphisms of almost complex manifolds, Ann. of Math., 77 (1963), 329–334.
Mathematical Reviews (MathSciNet): MR146764
Digital Object Identifier: doi:10.2307/1970219
JSTOR: links.jstor.org
T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631–661.
Mathematical Reviews (MathSciNet): MR998124
Zentralblatt MATH: 0850.70212
Digital Object Identifier: doi:10.2307/2001258
Jr. S. Gates, C. Hull, and M. Rocek, Twisted multiplets and new supersymmetric nonlinear $\sigma$-model, Nuclear Phys. B., 248(1) (1984), 157–186.
Mathematical Reviews (MathSciNet): MR776369
Digital Object Identifier: doi:10.1016/0550-3213(84)90592-3
V. Ginzburg, V. Guillemin, and Y. Karshon, Moment maps, cobordisms, and Hamiltonian group actions, Mathematical surveys and monographs, 98, Amer. Math. Soc., Providence, RI, 2002.
Mathematical Reviews (MathSciNet): MR1929136
Zentralblatt MATH: 1197.53002
V. Guillemin and S. Sternberg, Convexity properties of the moment mapping I and II, Invent. Math., 67 (1982), 491–513; Invent. Math., 77 (1984), 533–546.
Mathematical Reviews (MathSciNet): MR664117
Digital Object Identifier: doi:10.1007/BF01398933
M. Gualtieri, Generalized complex geometry, PhD thesis, Oxford University, 2004, math.DG/0401221.
N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math., 54 (2003), 281–308.
Mathematical Reviews (MathSciNet): MR2013140
Zentralblatt MATH: 1076.32019
Digital Object Identifier: doi:10.1093/qmath/hag025
A. Kapustin and A. Tomasiello, The general $(2, 2)$ gauged sigma model with three-form flux, preprint, hep-th/0610210.
Mathematical Reviews (MathSciNet): MR2362096
F. Kirwan, Convexity properties of the moment mapping III, Invent. Math., 77 (1984), 547–552.
Mathematical Reviews (MathSciNet): MR759257
Zentralblatt MATH: 0561.58016
Digital Object Identifier: doi:10.1007/BF01388838
E. Lerman, Symplectic cuts, Math. Res. Lett., 2 (1995), 247–258.
Mathematical Reviews (MathSciNet): MR1338784
Zentralblatt MATH: 0835.53034
E. Lerman, E. Meinrenken, S. Tolman, and C. Woodward, Non-abelian convexity by symplectic cuts, Topology, 37 (1998), 245–259.
Mathematical Reviews (MathSciNet): MR1489203
Digital Object Identifier: doi:10.1016/S0040-9383(97)00030-X
E. Lerman and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc., 349 (1997), 4201–4230.
Mathematical Reviews (MathSciNet): MR1401525
Zentralblatt MATH: 0897.58016
Digital Object Identifier: doi:10.1090/S0002-9947-97-01821-7
JSTOR: links.jstor.org
Y. Lin and S. Tolman, Symmetries in generalized Kähler geometry, Comm. Math. Phys., 268 (2006), 199–222.
Mathematical Reviews (MathSciNet): MR2249799
Digital Object Identifier: doi:10.1007/s00220-006-0096-z
Y. Lin, Generalized geometry, equivariant $\overline{\partial}\partial$-lemma, and torus actions, J. Geom. Phys., 57 (2007), 1842–1860.
Mathematical Reviews (MathSciNet): MR2330670
Zentralblatt MATH: 1170.53066
Digital Object Identifier: doi:10.1016/j.geomphys.2007.03.004
D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs, Oxford University Press, New York, 1995.
Mathematical Reviews (MathSciNet): MR1373431
I. Satake, The Gauss-Bonnet theorem for V-manifolds, J. Math. Soc. Japan, 9 (1957), 464–492.
Mathematical Reviews (MathSciNet): MR95520
Zentralblatt MATH: 0080.37403
Digital Object Identifier: doi:10.2969/jmsj/00940464
Project Euclid: euclid.jmsj/1261153826
Journal of the Mathematical Society of Japan
