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Rigid subsets of symplectic manifolds

Part of: Symplectic geometry, contact geometry

Published online by Cambridge University Press:  01 May 2009

Michael Entov  and
Leonid Polterovich
Michael Entov
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel (email: entov@math.technion.ac.il)
Leonid Polterovich
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (email: polterov@post.tau.ac.il)
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Abstract

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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 P. Biran-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.

Keywords

sympletic manifoldquantum homologyFloer homologyrigidity of intersectionsquasi-state

MSC classification

Secondary: 53D40: Floer homology and cohomology, symplectic aspects 53D12: Lagrangian submanifolds; Maslov index 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 53D20: Momentum maps; symplectic reduction

Information

Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 3 , May 2009 , pp. 773 - 826
Copyright
Copyright © Foundation Compositio Mathematica 2009

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