Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

By Michael Usher. 2010

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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;!/ 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;!/ to admit a Calabi quasimorphism and a symplectic quasistate. 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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d2fa06c897f963a60f3dad8cefa2c70fdc782355