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

SIGMA 9 (2013), 032, 33 pages      arXiv:1205.1870

On Orbifold Criteria for Symplectic Toric Quotients

Carla Farsi a, Hans-Christian Herbig b and Christopher Seaton c
a) Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, CO 80309-0395, USA
b) Centre for Quantum Geometry of Moduli Spaces, Ny Munkegade 118 Building 1530, 8000 Aarhus C, Denmark
c) Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, TN 38112, USA

Received August 07, 2012, in final form April 02, 2013; Published online April 12, 2013

We introduce the notion of regular symplectomorphism and graded regular symplectomorphism between singular phase spaces. Our main concern is to exhibit examples of unitary torus representations whose symplectic quotients cannot be graded regularly symplectomorphic to the quotient of a symplectic representation of a finite group, while the corresponding GIT quotients are smooth. Additionally, we relate the question of simplicialness of a torus representation to Gaussian elimination.

Key words: singular symplectic reduction; invariant theory; orbifold.

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  1. Barthel G., Brasselet J.P., Fieseler K.H., Kaup L., Combinatorial intersection cohomology for fans, Tohoku Math. J. 54 (2002), 1-41, math.AG/0002181.
  2. Bosio F., Meersseman L., Real quadrics in Cn, complex manifolds and convex polytopes, Acta Math. 197 (2006), 53-127, math.GT/0405075.
  3. Brøndsted A., An introduction to convex polytopes, Graduate Texts in Mathematics, Vol. 90, Springer-Verlag, New York, 1983.
  4. Cox D.A., Little J.B., Schenck H.K., Toric varieties, Graduate Studies in Mathematics, Vol. 124, American Mathematical Society, Providence, RI, 2011.
  5. Coxeter H.S.M., Regular complex polytopes, 2nd ed., Cambridge University Press, Cambridge, 1991.
  6. Cushman R., Sjamaar R., On singular reduction of Hamiltonian spaces, in Symplectic Geometry and Mathematical Physics (Aix-en-Provence, 1990), Progr. Math., Vol. 99, Birkhäuser Boston, Boston, MA, 1991, 114-128.
  7. Cushman R., Śniatycki J., Differential structure of orbit spaces, Canad. J. Math. 53 (2001), 715-755.
  8. Du Val P., Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
  9. Dunbar W.D., Greenwald S.J., McGowan J., Searle C., Diameters of 3-sphere quotients, Differential Geom. Appl. 27 (2009), 307-319, math.DG/0702680.
  10. Eisenbud D., The geometry of syzygies. A second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics, Vol. 229, Springer-Verlag, New York, 2005.
  11. Falbel E., Paupert J., Fundamental domains for finite subgroups in U(2) and configurations of Lagrangians, Geom. Dedicata 109 (2004), 221-238.
  12. Fulton W., Introduction to toric varieties, Annals of Mathematics Studies, Vol. 131, Princeton University Press, Princeton, NJ, 1993.
  13. Gessel I.M., Generating functions and generalized Dedekind sums, Electron. J. Combin. 4 (1997), no. 2, Paper 11, 17 pages.
  14. Gotay M.J., Bos L., Singular angular momentum mappings, J. Differential Geom. 24 (1986), 181-203.
  15. Haefliger A., Groupoïdes d'holonomie et classifiants, Astérisque (1984), no. 116, 70-97.
  16. Hatcher A., Vogtmann K., Rational homology of Aut(Fn), Math. Res. Lett. 5 (1998), 759-780.
  17. Herbig H.C., Iyengar S.B., Pflaum M.J., On the existence of star products on quotient spaces of linear Hamiltonian torus actions, Lett. Math. Phys. 89 (2009), 101-113, arXiv:0811.2152.
  18. Lerman E., Montgomery R., Sjamaar R., Examples of singular reduction, in Symplectic Geometry, London Math. Soc. Lecture Note Ser., Vol. 192, Cambridge University Press, Cambridge, 1993, 127-155.
  19. Marsden J., Weinstein A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121-130.
  20. Mather J.N., Differentiable invariants, Topology 16 (1977), 145-155.
  21. Meyer K.R., Symmetries and integrals in mechanics, in Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, 259-272.
  22. Molien T., Über die Invarianten der linearen Substitutionsgruppen, Sitzungsber. der Königl. Preuss. Akad. d. Wiss. (1897), zweiter Halbband, 1152-1156.
  23. Multarzyński P., Żekanowski Z., On general Hamiltonian dynamical systems in differential spaces, Demonstratio Math. 24 (1991), 539-555.
  24. Navarro González J.A., Sancho de Salas J.B., C-differentiable spaces, Lecture Notes in Mathematics, Vol. 1824, Springer-Verlag, Berlin, 2003.
  25. Pflaum M.J., Analytic and geometric study of stratified spaces, Lecture Notes in Mathematics, Vol. 1768, Springer-Verlag, Berlin, 2001.
  26. Schwarz G.W., Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63-68.
  27. Schwarz G.W., The topology of algebraic quotients, in Topological Methods in Algebraic Transformation Groups (New Brunswick, NJ, 1988), Progr. Math., Vol. 80, Birkhäuser Boston, Boston, MA, 1989, 135-151.
  28. Sikorski R., Wstęp do geometrii różniczkowej, Biblioteka Matematyczna, Vol. 42, Państwowe Wydawnictwo Naukowe, Warsaw, 1972.
  29. Sjamaar R., Lerman E., Stratified symplectic spaces and reduction, Ann. of Math. (2) 134 (1991), 375-422.
  30. Spanier E.H., Algebraic topology, Springer-Verlag, New York, 1981.
  31. Sturmfels B., Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993.

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