Abstract
In the first part of this paper we study different blow-upconstructions on symplectic orbifolds. Unlike the manifold case,we can define different blow-ups by using different circleactions. In the second part, we use some of these constructions todescribe the behavior of reduced spaces of a Hamiltonian circleaction on a symplectic orbifold, when passing a critical level ofits Hamiltonian function. Using these descriptions, we generalize,in the manifold case, the wall-crossing theorem of Guillemin and Sternberg to the case of a Hamiltonian torus action not necessarily quasi-free and also the Duistermaat–Heckman theorem to intervalsof values of the Hamiltonian function containing critical values.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Al Amrani, A.: Cohomological study of weighted projective spaces, In: S. Sertöz (ed.), Algebraic Geometry, Lecture Notes in Pure Appl. Math. 193, Springer-Verlag, New York, 1997, pp. 1-52.
Atiyah, M.: Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15.
Audin, M.: The Topology of Torus Actions on Symplectic Manifolds, Progr. in Math. 93, Birkhäuser, Boston, 1991.
Brion, M. and Procesi, C.: Action d'un tore dans une varieté projective, In: A. Connes, M. Duflo, A. Joseph and R. Rentschler (eds), Operator Algebras, Unitary Representations, Enveloping Algebras and Invariant Theory, Progr. in Math. 92, Birkhäuser, Boston, 1990, pp. 509-539.
Duistermaat, J. and Heckman, G.: On the variation in the cohomology of the symplectic form of the reduced phase, Invent. Math. 69 (1982), 259-269.
Fukaya, K. and Ono, K.: Arnold conjecture and Gromov-Witten invariants, Topology 38 (1999), 933-1048.
Guillemin, V.: Moment Maps and Combinatorial Invariants of Hamiltonian T n -Spaces, Progr. in Math. 122, Birkhäuser, Boston, 1994.
Guillemin, V. and Sternberg, S.: Convexity properties of the moment mapping I, Invent. Math. 67 (1982), 491-513.
Guillemin, V. and Sternberg, S.: Birational equivalence in the symplectic category, Invent. Math. 97 (1989), 485-522.
Guillemin, V. and Lerman, E.: Symplectic Fibrations and Multiplicity Diagrams, Cambridge Univ. Press, Cambridge, 1996.
Kawasaki, T.: Cohomology of twisted projective spaces and lens complexes, Math. Ann. 206 (1973), 243-248.
Kawasaki, T.: The signature theorem for V-manifolds, Topology 17 (1978), 75-83.
Lerman, E. and Tolman, S.: Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc. 349 (1997), 4201-4230.
McCarthy, J. and Wolfson, J.: Symplectic gluing along hypersurfaces and resolution of isolated orbifold singularities, Invent. Math. 119 (1995), 129-154.
McDuff, D. and Salamon, D.: Introduction to Symplectic Topology, Oxford Univ. Press, Oxford, 1995.
McDuff, D.: The moment map for circle actions on symplectic manifolds, J. Geom. Phys. 5 (1988), 149-160.
McDuff, D.: Almost complex structures on S 2 × S 2, Duke Math. J. 101 (2000), 135-177.
Roan, S.: Picard groups of hypersurfaces in toric varieties, Publ. RIMS, Kyoto Univ. 32 (1996), 797-834.
Satake, I.: On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. USA 42 (1956), 359-363
Satake, I.: The Gauss-Bonnet theorem for V-manifolds, J. Math. Soc. Japan 9 (1957), 464-492.
Sjamaar, R. and Lerman, E.: Stratified symplectic spaces and reduction, Ann. Math. 134 (1991), 375-422.
Thurston, W.: Three-Dimensional Geometry and Topology-Volume 1, Princeton Math. Series 35, Princeton Univ. Press, 1997.
Thurston, W.: Three-Dimensional Geometry and Topology-Volume 2, 2000, to appear.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Godinho, L. Blowing Up Symplectic Orbifolds. Annals of Global Analysis and Geometry 20, 117–162 (2001). https://doi.org/10.1023/A:1011628628835
Issue Date:
DOI: https://doi.org/10.1023/A:1011628628835