|
SIGMA 9 (2013), 032, 33 pages arXiv:1205.1870
https://doi.org/10.3842/SIGMA.2013.032
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
Abstract
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.
pdf (1061 kb)
tex (375 kb)
References
- 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.
- Bosio F., Meersseman L., Real quadrics in Cn, complex manifolds and
convex polytopes, Acta Math. 197 (2006), 53-127,
math.GT/0405075.
- Brøndsted A., An introduction to convex polytopes, Graduate Texts in
Mathematics, Vol. 90, Springer-Verlag, New York, 1983.
- Cox D.A., Little J.B., Schenck H.K., Toric varieties, Graduate Studies
in Mathematics, Vol. 124, American Mathematical Society, Providence, RI,
2011.
- Coxeter H.S.M., Regular complex polytopes, 2nd ed., Cambridge University Press,
Cambridge, 1991.
- 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.
- Cushman R., Śniatycki J., Differential structure of orbit spaces,
Canad. J. Math. 53 (2001), 715-755.
- Du Val P., Homographies, quaternions and rotations, Oxford Mathematical
Monographs, Clarendon Press, Oxford, 1964.
- 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.
- 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.
- Falbel E., Paupert J., Fundamental domains for finite subgroups in U(2) and
configurations of Lagrangians, Geom. Dedicata 109 (2004),
221-238.
- Fulton W., Introduction to toric varieties, Annals of Mathematics
Studies, Vol. 131, Princeton University Press, Princeton, NJ, 1993.
- Gessel I.M., Generating functions and generalized Dedekind sums,
Electron. J. Combin. 4 (1997), no. 2, Paper 11, 17 pages.
- Gotay M.J., Bos L., Singular angular momentum mappings, J. Differential
Geom. 24 (1986), 181-203.
- Haefliger A., Groupoïdes d'holonomie et classifiants, Astérisque
(1984), no. 116, 70-97.
- Hatcher A., Vogtmann K., Rational homology of Aut(Fn), Math.
Res. Lett. 5 (1998), 759-780.
- 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.
- 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.
- Marsden J., Weinstein A., Reduction of symplectic manifolds with symmetry,
Rep. Math. Phys. 5 (1974), 121-130.
- Mather J.N., Differentiable invariants, Topology 16 (1977),
145-155.
- Meyer K.R., Symmetries and integrals in mechanics, in Dynamical Systems
(Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New
York, 1973, 259-272.
- Molien T., Über die Invarianten der linearen Substitutionsgruppen,
Sitzungsber. der Königl. Preuss. Akad. d. Wiss.
(1897), zweiter Halbband, 1152-1156.
- Multarzyński P., Żekanowski Z., On general Hamiltonian dynamical
systems in differential spaces, Demonstratio Math. 24
(1991), 539-555.
- Navarro González J.A., Sancho de Salas J.B., C∞-differentiable
spaces, Lecture Notes in Mathematics, Vol. 1824, Springer-Verlag,
Berlin, 2003.
- Pflaum M.J., Analytic and geometric study of stratified spaces, Lecture
Notes in Mathematics, Vol. 1768, Springer-Verlag, Berlin, 2001.
- Schwarz G.W., Smooth functions invariant under the action of a compact Lie
group, Topology 14 (1975), 63-68.
- 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.
- Sikorski R., Wstęp do geometrii różniczkowej, Biblioteka Matematyczna, Vol. 42, Państwowe
Wydawnictwo Naukowe, Warsaw, 1972.
- Sjamaar R., Lerman E., Stratified symplectic spaces and reduction, Ann.
of Math. (2) 134 (1991), 375-422.
- Spanier E.H., Algebraic topology, Springer-Verlag, New York, 1981.
- Sturmfels B., Algorithms in invariant theory, Texts and Monographs in Symbolic
Computation, Springer-Verlag, Vienna, 1993.
|
|