Deformed Hamiltonian Floer theory, capacity estimates, and Calabi quasimorphisms

Abstract

We develop a family of deformations of the differential and of the pair-of-pants product on the Hamiltonian Floer complex of a symplectic manifold (M,ω) which upon passing to homology yields ring isomorphisms with the big quantum homology of M. By studying the properties of the resulting deformed version of the Oh-Schwarz spectral invariants, we obtain a Floer-theoretic interpretation of a result of Lu which bounds the Hofer-Zehnder capacity of M when M has a nonzero Gromov-Witten invariant with two point constraints, and we produce a new algebraic criterion for (M,ω) to admit a Calabi quasimorphism and a symplectic quasi-state. This latter criterion is found to hold whenever M has generically semisimple quantum homology in the sense considered by Dubrovin and Manin (this includes all compact toric M), and also whenever M is a point blowup of an arbitrary closed symplectic manifold.

73 pages, 1 color figure. Added new material explaining how to make some of the constructions on semipositive manifolds without using Kuranishi structures

References in corpus (1)

• Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle

Cited by in corpus (7)

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• Quasi-states, quasi-morphisms, and the moment map
• Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds
• Quantitative Heegaard Floer cohomology and the Calabi invariant
• Hofer-Zehnder capacity and Bruhat graph
• On the Hofer-Zehnder conjecture for semipositive symplectic manifolds