0705.3735May 25, 2007A minor change: clarified the recipe for computing the quantum homology of a symplectic toric Fano manifold

Symplectic quasi-states and semi-simplicity of quantum homology

Michael Entov, Leonid Polterovich

Symplectic Geometry Algebraic Geometry

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Abstract

We review and streamline our previous results and the results of Y.Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4-manifolds. We present also new results due to D.McDuff: she observed that for the existence of quasi-morphisms/quasi-states it suffices to assume that the quantum homology contains a field as a direct summand, and she showed that this weaker condition holds true for one point blow-ups of non-uniruled symplectic manifolds.

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