Nonrational, nonsimple convex polytopes in symplectic geometry
Authors:
Fiammetta Battaglia and Elisa Prato
Journal:
Electron. Res. Announc. Amer. Math. Soc. 8 (2002), 29-34
MSC (2000):
Primary 53D05; Secondary 53D20, 32S60, 52B20
DOI:
https://doi.org/10.1090/S1079-6762-02-00101-4
Published electronically:
September 17, 2002
MathSciNet review:
1928499
Full-text PDF Free Access
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Abstract: In this research announcement we associate to each convex polytope, possibly nonrational and nonsimple, a family of compact spaces that are stratified by quasifolds, i.e., the strata are locally modeled by ${\mathbb {R}}^k$ modulo the action of a discrete, possibly infinite, group. Each stratified space is endowed with a symplectic structure and a moment mapping having the property that its image gives the original polytope back. These spaces may be viewed as a natural generalization of symplectic toric varieties to the nonrational setting. We provide here the explicit construction of these spaces, and a thorough description of the stratification.
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[BBFK1]bbfk G. Barthel, J.-P. Brasselet, K.-H. Fieseler, L. Kaup, Equivariant intersection cohomology of toric varieties, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), 45–68, Contemp. Math. 241 Amer. Math. Soc., Providence, RI, 1999.
[BBFK2]bbfk2 G. Barthel, J.-P. Brasselet, K.-H. Fieseler, L. Kaup, Combinatorial intersection cohomology for fans, arXiv:math.AG/0002181.
[BP1]cx F. Battaglia, E. Prato, Generalized toric varieties for simple nonrational convex polytopes, Intern. Math. Res. Notices 24 (2001), 1315–1337.
[BP2]bp2 F. Battaglia, E. Prato, A symplectic realization of convex polytopes that are neither rational nor simple, in preparation.
[BL]bl P. Bressler, V. Lunts, Intersection cohomology on nonrational polytopes, arXiv: math.AG/0002006.
[D]delzant T. Delzant, Hamiltoniens périodiques et image convexe de l’application moment, Bull. Soc. Math. France 116 (1988), 315–339.
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[G]g V. Guillemin, Moment maps and combinatorial invariants of Hamiltonian $T^n$-spaces, Progress in Mathematics 122, Birkhäuser, Boston, 1994.
[LS]ls R. Sjamaar, E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math. 134 (1991), 375–422.
[P]p E. Prato, Simple nonrational convex polytopes via symplectic geometry, Topology 40 (2001), 961–975.
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Additional Information
Fiammetta Battaglia
Affiliation:
Dipartimento di Matematica Applicata “G. Sansone”, Via S. Marta 3, 50139 Firenze, Italy
Email:
fiamma@dma.unifi.it
Elisa Prato
Affiliation:
Laboratoire Dieudonné, Université de Nice, Parc Valrose, 06108 Nice Cedex 2, France
Email:
elisa@alum.mit.edu
Keywords:
Symplectic quasifolds,
moment mapping,
stratified spaces,
convex polytopes
Received by editor(s):
June 16, 2002
Published electronically:
September 17, 2002
Additional Notes:
The first author was partially supported by MIUR project Proprietà Geometriche delle Varietà Reali e Complesse, by GNSAGA (CNR), and by EDGE (EC FP5 Contract no. HPRN-CT-2000-00101).
Communicated by:
Frances Kirwan
Article copyright:
© Copyright 2002
American Mathematical Society
