Compositio Mathematica

Research Article

Rigid subsets of symplectic manifolds

Michael Entova1 and Leonid Polterovicha2

a1 Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel (email: entov@math.technion.ac.il)

a2 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (email: polterov@post.tau.ac.il)

Abstract

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.

(Received April 17 2007)

(Accepted July 14 2008)

(Online publication May 13 2009)

2000 Mathematics Subject Classification

  • 53D40;
  • 53D12;
  • 53D45;
  • 53D20

Keywords

  • sympletic manifold;
  • quantum homology;
  • Floer homology;
  • rigidity of intersections;
  • quasi-state

Footnotes

The first author was partly supported by E. and J. Bishop Research Fund and by the Israel Science Foundation grant # 881/06. Second author was partly supported by the Israel Science Foundation grant # 11/03.