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Reduced phase space and toric variety coordinatizations of Delzant spaces

Published online by Cambridge University Press:  01 May 2009

JOHANNES J. DUISTERMAAT  and
ALVARO PELAYO
JOHANNES J. DUISTERMAAT
Affiliation:
Mathematisch Instituut, Universiteit Utrecht, P.O. Box 80 010, 3508 TA Utrecht, The Netherlands. e-mail: [email protected]
ALVARO PELAYO
Affiliation:
University of California—Berkeley, Mathematics Department, 970 Evans Hall # 3840, Berkeley, CA 94720-3840, U.S.A. e-mail: [email protected]
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Abstract

In this note we describe the natural coordinatizations of a Delzant space defined as a reduced phase space (symplectic geometry view-point) and give explicit formulas for the coordinate transformations. For each fixed point of the torus action on the Delzant polytope, we have a maximal coordinatization of an open cell in the Delzant space which contains the fixed point. This cell is equal to the domain of definition of one of the natural coordinatizations of the Delzant space as a toric variety (complex algebraic geometry view-point), and we give an explicit formula for the toric variety coordinates in terms of the reduced phase space coordinates. We use considerations in the maximal coordinate neighborhoods to give simple proofs of some of the basic facts about the Delzant space, as a reduced phase space, and as a toric variety. These can be viewed as a first application of the coordinatizations, and serve to make the presentation more self-contained.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 3 , May 2009 , pp. 695 - 718
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[1]Abraham, R. and Marsden, J. E.Foundations of Mechanics (Benjamin/Cummings Publishing Co., 1978).Google Scholar
[2]Audin, M.The Topology of Torus Actions on Symplectic Manifolds (Birkhäuser, 1991).CrossRefGoogle Scholar
[3]Delzant, T.Hamiltoniens périodiques et images convexes de l'application moment. Bull. Soc. Math. France 116 (1988), 315–339.CrossRefGoogle Scholar
[4]Demazure, M.Sous–groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. École Norm. Sup. 3 (1970), 507–588.CrossRefGoogle Scholar
[5]Danilov, V. I.The geometry of toric varieties. Russ. Math. Surveys 33:2 (1978), 97–154, translated from Uspekhi Mat. Nauk SSSR 33:2 (1978), 85–134.CrossRefGoogle Scholar
[6]Duistermaat, J. J. and Heckman, G. J.On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69 (1982), 259–268.CrossRefGoogle Scholar
[7]Griffiths, P. and Harris, J.Principles of Algebraic Geometry (J. Wiley & Sons, Inc., 1978).Google Scholar
[8]Guillemin, V.Moment Maps and Combinatorial Invariants of Hamiltonian Tn–Spaces (Birkhäuser, 1994).CrossRefGoogle Scholar
[9]Guillemin, V.Kaehler structures on toric varieties. J. Differential Geom. 40 (1994), 285–309.CrossRefGoogle Scholar
[10]Hirzebruch, F.Über eine Klasse von einfach–zusammenhgenden komplexen Mannigfaltigkeiten. Math. Ann. 124 (1951), 77–86.CrossRefGoogle Scholar
[11]Pelayo, A.Topology of spaces of equivariant symplectic embeddings. Proc. Amer. Math. Soc. 135 (2007), 277–288.CrossRefGoogle Scholar
[12]Rockafellar, R. T.Convex Analysis (Princeton University Press, 1970).CrossRefGoogle Scholar