×

Rigid subsets of symplectic manifolds. (English) Zbl 1230.53080

Summary: We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P. Albers and of P. Biran and O. Cornea) as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
53D12 Lagrangian submanifolds; Maslov index
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
53D20 Momentum maps; symplectic reduction
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] doi:10.1215/00127094-2009-003 · Zbl 1183.53080 · doi:10.1215/00127094-2009-003
[5] doi:10.1007/BF01934324 · Zbl 0894.55006 · doi:10.1007/BF01934324
[6] doi:10.1007/s002200050268 · Zbl 0891.53022 · doi:10.1007/s002200050268
[7] doi:10.1112/blms/14.1.1 · Zbl 0482.58013 · doi:10.1112/blms/14.1.1
[12] doi:10.1007/BF01161996 · Zbl 0644.57024 · doi:10.1007/BF01161996
[13] doi:10.1007/BF01075861 · doi:10.1007/BF01075861
[15] doi:10.1016/j.aim.2003.09.009 · Zbl 1086.53067 · doi:10.1016/j.aim.2003.09.009
[16] doi:10.1002/cpa.20216 · Zbl 1142.53067 · doi:10.1002/cpa.20216
[17] doi:10.1007/BF01231767 · Zbl 0833.53029 · doi:10.1007/BF01231767
[19] doi:10.1155/IMRN.2005.2341 · Zbl 1126.53053 · doi:10.1155/IMRN.2005.2341
[21] doi:10.1007/BF01444643 · Zbl 0735.58019 · doi:10.1007/BF01444643
[22] doi:10.1016/0001-8708(91)90035-6 · Zbl 0744.46052 · doi:10.1016/0001-8708(91)90035-6
[24] doi:10.1112/S0010437X08003564 · Zbl 1151.53074 · doi:10.1112/S0010437X08003564
[26] doi:10.1007/BF01398933 · Zbl 0503.58017 · doi:10.1007/BF01398933
[27] doi:10.1007/BF01388438 · Zbl 0506.53030 · doi:10.1007/BF01388438
[29] doi:10.1007/BF01260388 · Zbl 0755.58022 · doi:10.1007/BF01260388
[30] doi:10.2140/pjm.2000.193.419 · Zbl 1023.57020 · doi:10.2140/pjm.2000.193.419
[35] doi:10.1002/cpa.3160451004 · Zbl 0766.58023 · doi:10.1002/cpa.3160451004
[36] doi:10.4171/CMH/43 · Zbl 1096.53052 · doi:10.4171/CMH/43
[38] doi:10.1155/S1073792803210011 · Zbl 1047.53055 · doi:10.1155/S1073792803210011
[41] doi:10.1016/0040-9383(93)90052-W · Zbl 0798.58018 · doi:10.1016/0040-9383(93)90052-W
[45] doi:10.1002/cpa.3160370204 · Zbl 0559.58019 · doi:10.1002/cpa.3160370204
[46] doi:10.1088/0951-7715/14/5/321 · Zbl 1067.37003 · doi:10.1088/0951-7715/14/5/321
[47] doi:10.1016/j.geomphys.2008.06.003 · Zbl 1161.53076 · doi:10.1016/j.geomphys.2008.06.003
[48] doi:10.1155/S1073792898000178 · Zbl 0939.37030 · doi:10.1155/S1073792898000178
[49] doi:10.1155/S1073792804132716 · Zbl 1079.53133 · doi:10.1155/S1073792804132716
[53] doi:10.1007/s00039-006-0560-0 · Zbl 1099.53054 · doi:10.1007/s00039-006-0560-0
[54] doi:10.2140/agt.2006.6.405 · Zbl 1114.53070 · doi:10.2140/agt.2006.6.405
[58] doi:10.1002/cpa.3160481104 · Zbl 0847.58036 · doi:10.1002/cpa.3160481104
[59] doi:10.1142/S0219199704001525 · Zbl 1076.53110 · doi:10.1142/S0219199704001525
[61] doi:10.1007/s00014-001-8326-7 · Zbl 1001.53057 · doi:10.1007/s00014-001-8326-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.