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Rigid subsets of symplectic manifolds

Published online by Cambridge University Press:  01 May 2009

Michael Entov
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel (email: [email protected])
Leonid Polterovich
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (email: [email protected])
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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.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2009

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