Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

Usher, Michael¹

¹ Department of Mathematics, University of Georgia, Athens GA 30602, USA

Geometry & topology, Tome 15 (2011) no. 3, pp. 1313-1417.

Résumé

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 $\left(M,\omega \right)$ 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 $\left(M,\omega \right)$ 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.

Détail

DOI : 10.2140/gt.2011.15.1313

Classification : 53D40, 53D45
Keywords: Hamiltonian Floer theory, spectral invariant, quasimorphism, semisimple quantum homology

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